A    TEXT-BOOK 


ON 


HYDRAULICS 


INCLUDING   AN    OUTLINE   OF   THE 


THEORY    OF    TURBINES 


BY 


L.    M.    HOSKINS 

' 

Professor  of  Applied  Mathematics  in  the 
Leland  Stanford  Junior  University 


NEW    YORK 

HENRY    HOLT    AND    COMPANY 

1907 


Copyright,  1906, 

BY 
HENRY  HOLT  AND  COMPANY 


ROBERT  DRCMMOND,  PRINTER.  NEW  YORK 


PEEFACE, 

THIS  book  is  designed  primarily  for  the  use  of  students  of 
engineering  in  universities  and  technical  colleges.  In  its  prep- 
aration the  aim  has  been  to  present  fundamental  principles  in 
a  manner  both  sound  and  as  simple  as  possible.  The  treatment 
presupposes  a  good  elementary  knowledge  of  the  principles  of 
mechanics,  and  a  working  knowledge  of  the  elements  of  cal- 
culus; but  to  the  student  thus  equipped,  who  is  also  well  trained 
in  arithmetic,  algebra  and  trigonometry,  it  presents  little  mathe- 
matical difficulty.  Many  numerical  examples  Are  introduced, 
the  complete  solution  of  which  should  form  an  important  part 
of  the  work  of  the  student. 

It  is  perhaps  not  too  much  to  say  that  the  key  to  a  correct 
understanding  of  all  problems  in  the  steady  flow  of  liquids  is 
supplied  by  Bernoulli's  theorem, — or,  as  it  is  usually  called  in 
the  text,  the  general  equation  of  energy.  Familiarity  with  this 
principle  is  therefore  much  more  important  than  a  memory- 
knowledge  of  special  rules,  and  for  this  reason  the  explanations 
of  particular  cases  of  flow  have  in  most  cases  been  based  directly 
upon  the  fundamental  equation.  The  meaning  and  importance 
of  the  term  representing  lost  energy  in  this  equation  have  also 
been  emphasized.  The  corresponding  theory  applied  to  gases 
is  given  in  Appendix  A. 

In  the  presentation  of  working  rules  for  estimating  flow  in 
the  various  practical  cases  met  by  the  engineer  it  has  been 
aimed  in  every  case  to  give  a  clear  statement  of  the  rational 
basis  of  the  formula  adopted,  and  also  to  make  clear  to  what 
extent  the  theory  is  defective  and  the  formula  therefore  empiri- 
cal. It  has  also  been  attempted  to  avoid  the  appearance  of 

in 

161891 


iv  PREFACE. 

precise  knowledge  where  the  reality  is  absent.  For  example, 
no  elaborate  tables  have  been  given  purporting  to  show  accu- 
rately how  frictional  loss  of  head  in  pipes  depends  upon  velocity 
and  diameter,  or  giving  precise  values  of  the  friction  factor  for 
pipes  of  different  kinds.  A  somewhat  careful  study  of  experi- 
mental data  has  failed  to  convince  the  author  of  the  reliability 
of  any  such  tables. 

The  treatment  of  turbines  and  water  wheels  has  been  re- 
stricted to  an  outline  of  the  theory,  but  several  illustrations 
showing  typical  American  practice  have  been  included.  For 
these  the  author  is  indebted  to  the  courtesy  of  manufacturers, 
to  whom  credit  is  in  every  case  given  in  the  text.  The  aim  has 
been  to  unify  the  theory,  the  treatment  of  all  specific  cases 
being  based  upon  the  same  general  principles  and  equations, 
and  a  general  notation  for  velocities  and  their  direction-angles 
being  adopted  which  it  is  hoped  will  be  found  simple  and  helpful. 
This  unification  includes  the  theory  of  turbine  pumps. 

In  various  discussions  throughout  the  book  reference  is 
made  to  the  author's  text-book  on  Theoretical  Mechanics  for  a 
fuller  explanation  of  basal  principles. 

L.  M.  H. 

PALO  ALTO,  CAL.,  June,  1906. 


CONTENTS. 


CHAPTER  PAGE 

I.  PRELIMINARY  DEFINITIONS  AND  PRINCIPLES 1 

II.  HYDROSTATICS . 7 

III.  FLOW   OF   WATER  THROUGH   ORIFICES.    TORRICELLI'S   THEO- 

REM   30 

IV.  THEORY  OF  ENERGY  APPLIED  TO  STEADY  STREAM  MOTION.  ...  43 
V.  APPLICATION  OF  GENERAL  EQUATION  OF  ENERGY,  NEGLECTING 

LOSSES  BY  DISSIPATION 52 

VI.  APPLICATION  OF  GENERAL  EQUATION  OF  ENERGY,  TAKING  AC- 
COUNT OF  LOSSES 64 

VII.  GENERAL  EQUATION  OF   ENERGY  WHEN  PUMP  OR  MOTOR  is 

USED 85 

VIII.  FLOW  IN  PIPES:  SPECIAL- CASES 90 

IX.  FRICTIONAL  Loss  OF  HEAD  IN  PIPES 102 

X.  EQUATION  OF  ENERGY  FOR  STREAM  OF  LARGE  CROSS-SECTION.  116 

XI.  UNIFORM  FLOW  IN  OPEN  CHANNELS 122 

XII.  OPEN  CHANNELS:  NON-UNIFORM  FLOW. 135 

XIII.  THE  MEASUREMENT  OF  RATE  OF  DISCHARGE 142 

XIV.  DYNAMIC  ACTION  OF  STREAMS 160 

XV.  THEORY  OF  STEADY  FLOW  THROUGH  ROTATING  WHEEL 176 

XVI.  TYPES  OF  TURBINES  AND  WATER  WHEELS 185 

XVII.  THEORY  OF  IMPULSE  TURBINE 190 

XVIII.  THE  TANGENTIAL  WATER  WHEEL 199 

XIX.  THEORY  OF  THE  REACTION  TURBINE 208 

XX.  TURBINE  PUMPS 224 

APPENDIX  A.     STEADY  FLOW  OF  A  GAS 249 

APPENDIX  B.     RELATIVE  MOTION 261 

APPENDIX  C.     CONVERSION  FACTORS 264 

INDEX 267 

v 


XA; 

I  UNIVERSITY  1 

>K  r\c 

^^ 


HYDRAULICS 

CHAPTER  I. 
PRELIMINARY  DEFINITIONS  AND  PRINCIPLES. 

1.  Definition  of  Subject. — The  mechanics  of  fluid  bodies  is 
called  Hydromechanics.     It  embraces  Hydrostatics,  dealing  with 
the  principles  of  fluid  equilibrium,  and  Hydrokinetics,  dealing 
with  the  laws  of  fluid  motion. 

Hydraulics,  the  subject  of  this  book,  may  be  defined  briefly 
as  practical  Hydromechanics.  It  deals  especially  with  the  flow 
of  water  in  streams  of  various  kinds,  but  may  be  taken  to  in- 
clude all  the  principles  and  applications  of  Hydromechanics 
that  bear  directly  upon  problems  of  practical  utility.  Many 
of  the  laws  of  Hydraulics  are  largely  empirical,  but  certain 
fundamental  dynamical  principles,  especially  the  law  of  energy, 
serve  to  unify  the  subject  and  to  put  it  upon  a  scientific  basis. 

The  bodies  dealt  with  in  Hydromechanics  may  be  either 
liquids  or  gases,  Hydraulics  deals  mainly,  but  not  exclusively, 
with  liquids,  and  especially  with  water. 

2.  Distinction  between  Solid  and  Fluid  Bodies. — A  solid  body 
can  permanently  resist  change  of  shape;  a  fluid  body  cannot. 

A  fluid  is  either  liquid  or  gaseous.  A  gas  tends  to  expand 
indefinitely,  so  as  to  fill  any  continuous  closed  volume  in  any 
portion  of  which  it  may  be  placed.  A  liquid  changes  its  vol- 
ume only  slightly  under  changes  of  pressure;  a  given  portion 
may  be  wholly  freed  from  external  pressure  without  expanding 
beyond  a  certain  volume. 


PRELIMINARY  DEFINITIONS  AND  PRINCIPLES. 

The  distinction  between  solids  and  fluids  may  be  made  more 
definite  by  a  consideration  of  internal  stresses. 

3.  Internal  Stresses  in  a  Body. — The  two  equal  and  oppo- 
site forces  exerted  by  two  portions  of  matter  upon  each  other 
constitute  a  stress.    The  forces  making  up  a  stress  are  thus  the 
"action  and  reaction  "  of  Newton's  third  law  of  motion. 

If  the  two  portions  of  matter  are  parts  of  the  same  body, 
the  stress  is  internal  with  reference  to  that  body.  Internal 
stresses,  acting  between  adjacent  portions  of  a  body,  are  called 
into  action  whenever  external  forces  tend  to  change  the  shape 
or  size  of  the  body. 

If  a  plane  surface  be  conceived  to  divide  a  body  into  two 
contiguous  parts,  the  forces  which  these  parts  exert  upon  each 
other  will  for  convenience  be  regarded  as  resolved  into  com- 
ponents normal  and  tangential  to  the  plane.  The  stress  com- 
posed of  these  forces  is  thus  resolved  into 

(a)  A  normal  stress,  which  resists  whatever  tendency  there 
may  be  for  the  two  parts  of  the  body  to  approach  or  recede 
from  each  other  in  the  direction  of  the  normal  to  the  plane  of 
separation,  and 

(6)  A  tangential  stress,  which  resists  any  tendency  to  slid- 
ing, or  relative  motion  parallel  to  the  plane. 

4.  Mathematical   Definitions   of   Solid   and  Fluid. — A    fluid 
body  is  one  in  which  tangential  stress  cannot  act,  except  while 
the  shape  of  the  body  is  changing. 

If,  by  reason  of  external  forces,  any  two  adjacent  portions 
of  a  fluid  tend  to  slide  over  each  other,  tangential  stresses 
come  into  action  to  resist  such  sliding.  The  sliding  is  not  pre- 
vented, however,  but  continues  until  a  condition  of  equilibrium 
is  attained;  in  this  condition  the  tangential  stress  on  every 
plane  vanishes. 

A  solid  body  is  one  in  which  tangential  stresses  can  act 
permanently  to  resist  change  of  shape. 

A  " perfect"  fluid  may  be  defined  as  one  which  offers  no 
resistance  to  change  of  shape.  In  other  words,  no  tangential 


PRESSURE;  INTENSITY  OF  PRESSURE.         3 

stress  acts  in  a  perfect  fluid  even  while  the  particles  are  sliding 
over  one  another.     No  known  fluid  is  perfect  in  this  sense. 

The  laws  of  equilibrium  are  the  same  for  an  actual  fluid  as 
for  a  perfect  fluid,  since  it  is  only  when  the  parts  of  a  body  of 
fluid  move  relatively  to  one  another  that  tangential  stresses 
act.  In  other  words,  the  statics  of  actual  fluids  is  the  same  as 
the  statics  of  perfect  fluids. 

5.  Pressure  ;  Intensity  of  Pressure.  —  The  normal  stress  be- 
tween two  adjacent  parts  of  a  body  may  be  either  tensile  or 
compressive. 

Tensile  stress  (or  tension)  resists  a  tendency  of  the  two  por- 
tions of  the  body  to  separate. 

Compressive  stress  (or  pressure)  resists  a  tendency  of  the  two 
portions  of  the  body  to  approach  each  other. 

In  Hydromechanics  we  are  concerned  mainly  with  pressure, 
since  a  fluid  body  can  sustain  only  a  slight  tensile  stress.  We 
shall  have  to  consider  not  only  internal  pressure,  but  also  pres- 
sure acting  between  a  body  of  water  and  other  bodies  in  con- 
tact with  it. 

Intensity  of  pressure  means  pressure  per  unit  area.  If,  on 
any  surface  subject  to  pressure,  the  pressures  upon  any  two 
elementary  areas,  however  small,  are  proportional  to  the  areas, 
the  intensity  of  pressure  is  uniform  over  the  surface.  In  this 
case  its  value  is  at  every  point  equal  to  the  total  pressure 
divided  by  the  total  area.  Algebraically,  let 

P  =  total  pressure  on  area  F\ 

p=  intensity  of  pressure  at  any  point  of  the  area; 

then  P 


If  the  intensity  of  pressure  has  not  the  same  value  at  all 
points  of  the  area,  we  may  regard  P/F  as  its  average  value 
for  the  area  F.  The  true  value  of  p  at  any  given  point  may 
be  expressed  approximately  as  its  average  value  over  a  small 
area  containing  the  point.  If  4F  is  the  area  of  this  element 


PRELIMINARY  DEFINITIONS  AND  PRINCIPLES. 

and  AP  the  total  pressure  on  the  element,  we  have  approxi- 
mately 

*£ 

p         ' 


the  approximation  being  closer  the  smaller  the  element. 

Taking  AF  smaller  and  smaller  with  limit  0,  but  always 
containing  the  point  at  which  the  value  of  p  is  to  be  expressed, 
we  have  as  the  exact  value  of  the  intensity  of  pressure  at  that 
point 

AP    dP 


For  brevity  the  word  "  pressure  "  is  often  used  instead  of 
"  intensity  of  pressure."  This  abbreviation  will  sometimes  be 
employed  in  the  following  discussions,  but  should  be  avoided 
when  it  is  liable  to  cause  ambiguity. 

Although  the  foregoing  discussion  of  intensity  of  stress  has 
referred  to  normal  stresses  only,  similar  considerations  hold 
for  tangential  stresses. 

6.  Elasticity.—  Elasticity  is  the  property  by  virtue  of  which 
a  body  regains  its  original  size  and  shape  (in  whole  or  in  part) 
after  these  have  been  changed  by  the  action  of  external  forces. 

Both  elasticity  of  volume  and  elasticity  of  shape  are  pos- 
sessed in  very  different  degrees  by  different  bodies.  Fluid 
bodies  possess  practically  perfect  elasticity  of  volume,  but  no 
elasticity  of  shape. 

Since  change  of  shape  always  involves  sliding,  or  tangential 
motion  of  the  parts  of  a  body  relative  to  one  another,  the  body 
cannot  of  itself  regain  its*  original-  shape  except  by  the  con- 
tinued action  of  the  tangential  stresses  which  resist  such  motion. 
A  fluid  body,  having  been  deformed  from  one  form  of  equilib- 
rium to  another,  cannot  of  itself  return  to  the  original  shape, 
because  after  equilibrium  is  attained  no  tangential  stresses  are 
in  action. 


DENSITY  AND  COMPRESSIBILITY  OF  WATER. 


7.  External  Forces. — A  force  acting  upon  any  portion  of  a 
body  is  called  external  if  the  portion  of  matter  exerting  the 
force  is  not  a  part  of  the  body. 

The  external  forces  acting  upon  any  body  of  fluid  are  of  two 
classes:  (a)  surface  forces  and  (6)  bodily  forces. 

(a)  A  surface  force  is  one  whose  place  of  application  is  some 
portion  of  the  bounding  surface  of  the  body.  Such  forces  are 
exerted  by  other  bodies  in  contact  with  the  given  body. 

(6)  A  bodily  force  is  one  which  is  applied  throughout  some 
definite  volume  of  the  body.  Such  a  force  does  not  depend 
upon  contact  between  the  two  bodies  concerned;  it  is  of  the 
kind  called  "action  at  a  distance."  .An  example  of  a  bodily 
force  is  gravity,  acting  upon  every  particle  of  a  body  of  fluid 
near  the  earth's  surface.  In  practical  Hydraulics  this  is  the 
only  bodily  force  to  be  considered. 

8.  Density  and  Compressibility  of  Water.  —  Practical   Hy- 
draulics deals  mainly  with  water.    The  properties  of  water  with 
which  we  shall  chiefly  be  concerned  are  its  density  and  com- 
pressibility. 

Density. — The  density  (or  mass  per  unit  volume)  of  water 
varies  but  little  with  pressure  and  temperature  within  ordinary 
ranges.  The  density  of  pure  water  at  several  different  tem- 
peratures and  under  ordinary  atmospheric  pressure  is  given  in 
the  following  table: 

TABLE  I. 


Temp,  Fahr. 

Density  in  Ibs. 
per  cu.  ft. 

Temp.  Cent. 

Density  in  kgr. 
per  cu.  met. 

32° 

62.42 

0° 

999.9 

39.3 

62.424 

4 

1000.0 

50 

62.41 

10 

999.8 

60 

62.37 

15 

999.2 

70 

62.30 

20 

998.3 

80 

62.22 

25 

997.1 

90 

62.12 

30 

995.8 

100 

62.00 

35 

994.6 

110 

61.86 

40 

992.4 

The  density  of  ordinary  terrestrial  water  is  increased  very 
slightly  by  the  presence  of  various  substances  in  solution.     In 


6  PRELIMINARY  DEFINITIONS  AND  PRINCIPLES. 

most  ordinary  computations  the  convenient  numbers  62.5  Ibs. 
per  cubic  foot  and  1000  kilograms  per  cubic  meter  may  be  em- 
ployed; 62.4  Ibs.  per  cubic  foot  is,  however,  more  accurate 
than  the  former  value. 

The  density  of  sea- water  is  about  2.6  per  cent  greater  than 
that  of  pure  water. 

Compressibility. — The  compressibility  of  water  is  so  small 
that  for  ordinary  purposes  it  may  be  neglected.  The  ratio  to 
the  original  volume  of  the  decrease  in  volume  caused  by  a  given 
pressure  (of  uniform  intensity  over  the  whole  bounding  surface) 
may  be  taken  as  a  measure  of  the  compressibility.  The  value 
of  this  ratio  for  a  pressure  of  1  atmosphere  *  varies  somewhat 
with  the  temperature.  Near  the  freezing-point  it  is  about 
0.000050;  at  77°  Fahr.  (25°  Cent.)  it  is  about  0.000045. 

*  See  Art.  14. 


CHAPTER  II. 
HYDROSTATICS. 

9.  Pressure  on  Different  Planes  Passing  Through  a  Point.  — 

From  the  fact  that  no  tangential*  stress  exists  on  any  plane  in  a 
body  of  fluid  in  equilibrium,  it  follows  that  the  intensity  of 
normal  pressure  has,  at  a  given  point,  the  same  value  for  all 
planes  passing  through  that  point. 

Consider  any  two  planes  passing  through  the  point  0 
(Fig.  1).  Let  OM,  ON  be  their  traces  on  a  plane  perpendicular 
to  both  which  is  taken  as  the  plane 
of  the  figure.  Take  OA  =  OB,  and  \  / 

let  AB  be  the  trace  of   a   third         \  _  / 
plane  which   is   perpendicular   to  \        /w,  x*z  sina  cosa 


the  plane  of  the  figure.    Consider 

the  body  of  fluid  bounded  by  five 

planes  as  follows:    The  plane  of 

the  figure  ;  a  plane  parallel  to  it  at 

a  distance  2;  planes  perpendicular 

to  the  plane  of  the  figure  having  traces  OA,  OB,  AB.    The  forces 

acting  upon  this  body  are  the  normal  pressures  exerted  upon 

every  part  of  its  bounding  surface  by  the  surrounding  fluid,  and 

whatever  bodily  forces  may  be  acting  upon  it. 

Let  the  average  value  of  the  bodily  force  per  unit  volume, 
for  the  body  just  described,  be  w',  and  let 

Wi  =<  component  of  w'  in  direction  AB', 

pi  =  average  intensity  of  pressure  on  the  face  OA; 

p2  =  average  intensity  of  pressure  on  the  face  OB; 

2a=  angle  AOB; 

x  =  OA=OB. 

7 


8  HYDROSTATICS. 

Then  pixz  =  total  pressure  acting  on  face  OA ; 

p2xz=    "          "          "      "     "    OB; 
WiX2z  sin  a  cos  a  =  total  bodily  force  in  direction  AB. 

These  are  the  only  forces  acting  on  the  body  which  are 
not  perpendicular  to  AB;  hence  for  equilibrium,  resolving  in 
direction  AB, 

PIXZ  cos  a  —p2xz  cos  a  +WiX2z  sin  a  cos  a  =0, 
or  pi—  p2+WiX  sin  a  =  0. 

This  equation  is  true  for  any  values  of  x  and  z.  If  both  be 
made  to  approach  0,  the  third  term  of  the  equation  approaches 
0,  so  that 

limit  pi  =  limit  p2. 

But  the  limiting  values  of  pi  and  p2  are  the  true  values  of  the 
intensity  of  pressure  at  the  point  0  on  the  planes  OA  and  OB 
respectively.  And  since  these  may  be  any  two  planes  passing 
through  0  without  changing  the  reasoning,  the  proposition  is 
established. 

10.  Normal  Pressure  in  a  Fluid  Free  from  Bodily  Forces. — 

Although  in  all  the  practical  problems  of  Hydraulics  and  Hydro- 
statics the  fluids  considered  are  acted  upon  by  the  force  of 
gravity,  it  is  instructive  to  consider  the  ideal  case  of  a  fluid  not 
acted  upon  by  any  bodily  force. 

It  is  easily  shown  that  for  such  a  body  in  equilibrium  the 
intensity  of  pressure  has  the  same  value  at  all  points. 

Let  A  and  B  (Fig.  2)   be  any  two  points  such  that  the 
straight    line    AB    lies    wholly   within    the   fluid.       Consider 
an  elementary  prism  of  small    cross-section    F,    so 
taken  that  A  and  B  lie  in  its  two    bases.     Let  pi 
and  p2  denote  the  values  of  the  intensity  of  pressure 
at  A  and  B  respectively,  and  let  all  forces  acting 
upon  the  prismatic  element  be  resolved  parallel  to 
FIG.  2.    its  axis.    The  only  forces  not  normal  to  the  direc- 
tion of  resolution  are  the  pressures  on  the  end  elements  at  A 
and  B;  hence  for  equilibrium  we  have 

p2F  =  0,     or    pi  —  p2. 


PRESSURE  IN  FLUID  ACTED   UPON   BY  GRAVITY.  9 

That  is,  the  intensity  of  pressure  has  the  same  value  at  A  and 
at  B.  This  result  may  obviously  be  extended  to  any  two 
points  in  a  continuous  body  of  fluid  free  from  the  action  of 
bodily  forces. 

11.  Variation  of  Pressure  in  Fluid  Acted  upon  by  Gravity.— 

In  case  of  fluids  at  the  earth's  surface  the  only  bodily  force  to 
be  considered  is  the  attraction  of  the  earth  upon  every  particle. 
For  a  fluid  of  uniform  density  this  attraction  (per  unit  volume) 
has  practically  the  same  magnitude  and  direction  at  all  points. 
If  the  weight  of  a  pound  mass  (called  a  pound  force)  is  taken  as 
the  unit  force,  the  force  per  unit  volume  has  the  same  value 
as  the  mass  (in  pounds)  per  unit  volume. 

The  law  of  variation  of  pressure  in  a  body  of  fluid  acted 
upon  by  gravity  may  be  determined  as  follows: 

Let  A  and  B  (Fig.  3)  be  any  two  points  such  that  the 
straight   line   AB  lies  wholly  in  the  fluid. 
Let  BC  be  horizontal  and  AC  vertical,  and 
let  AB  =  l,  AC=z. 

Consider  an  elementary  right  prism  with 
axis  AB  and  cross-section  F,  the  bases  con- 
taining the  points  A  and  B.  Let  all  forces 
acting  upon  this  prism  be  resolved  parallel 
to  AB.  Let 

pi  =  intensity  of  pressure  at  A ; 
p2=  in  tensity  of  pressure  at  B, 

w  =  weight  of  fluid  per  unit  volume; 

a  =  angle  BAG. 

For  equilibrium, 

plF-p2F+wFlcosa=Q. 

Or,  since  I  cos  a  =  z, 

9 

p2-pi=wz. 

This  result  may  be  extended  to  any  two  points  of  a  con- 
nected fluid,  and  may  be  stated  in  general  terms  as  follows: 
In  any  body  of  homogeneous  fluid  in  equilibrium  the  pres- 


10  HYDROSTATICS. 

sure  varies  in  direct  ratio  with  the  depth.  The  difference 
between  the  values  of  the  pressure  (per  unit  area)  at  any  two 
points  is  equal  to  the  weight  of  a  prism  of  the  fluid  of  unit 
cross-section  and  of  length  equal  to  the  vertical  distance  be- 
tween the  two  points. 

The  following  discussions  will  refer  usually  to  the  case  of  a 
liquid  acted  upon  by  gravity. 

Units. — It  is  to  be  remembered  that  a  consistent  system  of 
units  must  be  used  in  applying  the  above  formula.  Through- 
out this  book  the  foot  will  nearly  always  be  used  as  the  unit 
length,  and  the  pound  as  the  unit  force.  The  value  of  w  is 
therefore  62.5  (more  accurately  62.4)  pounds  per  cubic  foot, 
z  is  to  be  expressed  in  feet,  and  p  in  pounds  per  square  foot. 
With  French  units,  if  the  meter  is  taken  as  the  unit  length 
and  the  kilogram  as  the  unit  force,  the  value  of  w  is  the  weight 
in  kilograms  of  a  cubic  meter  of  water,  or  1,000. 

12.  Surface  of  Equal  Pressure.— As  a  special  case  of  the 
above  proposition  it  is  seen  that  if  the  intensity  of  pressure 
has  the  same  value  at  any  two  points,  these  must  lie  in  the 
same  horizontal  plane.     Any  horizontal  plane  is,  in  fact,  a 
surface  of  equal  pressure. 

A  surface  of  equal  pressure  is  called  a  level  surface. 

Free  surface.— The  bounding  surface  of  a  liquid  is  said  to 
be  free  if  not  sustaining  pressure.  Ordinarily  the  upper  sur- 
face of  a  body  of  water  is  called  free  although  under  atmos- 
pheric pressure.  Whether  under  zero  pressure  or  any  uniform 
pressure,  the  free  surface  of  a  liquid  in  equilibrium  is  obviously 
a  horizontal  plane  *  if  gravity  is  the  only  bodily  force. 

13.  Pressure  Expressed  in  Terms  of  Height  of  Liquid  Column. 

— Fluid  pressure  is  often  estimated  in  terms  of  the  "  equivalent 
height"  of  some  specified  liquid.  Thus,  a  column  of  water  1 
foot  high  is  said  to  be  "  equivalent  to  "  a  pressure  of  about 

*  The  accurate  statement  is  that  the  free  surface  (or  any  level  surface) 
is  everywhere  normal  to  the  direction  of  gravity,  and  is  approximately  a 
spherical  surface  concentric  with  the  earth. 


ATMOSPHERIC  PRESSURE.  11 

62.5  pounds  per  square  foot,  since  the  intensity  of  pressure  at 
the  base  of  such  a  column  exceeds  that  at  the  top  by  the 
amount  stated.  In  discussions  in  Hydraulics  it  is  quite  com- 
mon to  express  pressures  in  this  way. 

In  scientific  investigations  mercury  is  often  taken  as  the 
standard  fluid,  pressures  being  expressed  as  so  many  inches, 
or  centimeters,  of  mercury. 

Since  the  density  of  mercury  is  about  13.6  times  that  of 
water,  the  height  of  the  water  column  equivalent  to  a  given 
pressure  is  about  13.6  times  as  great  as  that  of  the  correspond- 
ing mercury  column. 

14.  Atmospheric  Pressure. — The  air  exerts  upon  all  terres- 
trial bodies  a  pressure  whose  intensity  is  equal  to  the  weight 
of  a  column  of  air  of  unit  cross-section  extending  upward  com- 
pletely through  the  atmosphere.  In  many  hydraulic  problems 
this  pressure  may  be  disregarded  because  its  effects  at  different 
points  counterbalance.  It  is  quite  common  to  reckon  pressures 
from  atmospheric  pressure  as  zero,  pressures  of  less  intensity 
being  regarded  as  negative.  An  actual  negative  pressure  (i.e., 
a  tensile  stress)  of  any  considerable  intensity  cannot  exist  in 
a  liquid,  any  tendency  to  such  a  stress  resulting  in  a  separation 
of  the  parts  of  the  liquid. 

The  intensity  of  atmospheric  pressure  at  any  locality  varies 
somewhat  with  weather  conditions,  and  at  different  places  it 
varies  with  the  elevation.  At  sea-level  under  ordinary  condi- 
tions the  value  is  about  14.72  pounds  per  square  inch,  or  2120 
pounds  per  square  foot. 

The  height  of  the  equivalent  water  column  is  very  nearly 
34  feet  (10.34  meters),  and  that  of  the  equivalent  mercury 
column  30  inches  (76  centimeters). 

At  any  height  above  sea-level  the  corresponding  values 
may  be  found  by  multiplying  the  above  numbers  by  the 
proper  factor  taken  from  the  following  table,  which  gives 
the  ratio  of  atmospheric  pressure  at  any  elevation  to  its 
value  at  sea-level.  It  must  be  understood  that  the  results 
will  be  only  approximate,  the  table  being  computed  for  the 


12 


HYDROSTATICS. 


ideal  case   of   a  perfect  gas  at  the   uniform  temperature   of 
0°  Cent. 

TABLE  IT. 


Elevation  above 
sea-level  in  feet. 

Ratio  of  atmos- 
pheric pressure  to 
its  value  at  sea- 
level. 

Elevation  above 
sea-level  in  feet. 

Ratio  of  atmos- 
pheric pressure  to 
its  value  at  sea- 
level. 

500 

.9811 

8,000 

.7370 

1,000 

.9626 

8,500 

.7231 

1,500 

.9444 

9,000 

.7094 

2,000 

.9265 

9,500 

.6960 

2,500 

.9090 

10,000 

.6829 

3,000 

.8918 

10,500 

.6699 

3,500 

.8750 

11,000 

.6573 

4,000 

.8584 

11,500 

.6449 

4,500 

.8422 

12,000 

.6327 

5,000 

.8263 

12,500 

.6207 

5,500 

.8107 

13,000 

.6090 

6,000 

.7954 

13,500 

.5975 

6,500 

.7804 

14,000 

.5862 

7,000 

.7656 

14,500 

.5751 

7,500 

.7512 

15,000 

.5642 

15.  Resultant  Pressure. — The  resultant  pressure  on  any  sur- 
face, plane  or  curved,  is  found  by  combining  the  pressures  on 
the  elementary  portions  of  the  surface,  having  regard  for  direc- 
tion as  well  as  magnitude. 

The  point  of  application  of  the  resultant  pressure,  for  a 
given  surface,  is  called  the  center  of  pressure. 

16.  Resultant  Pressure    on    Horizontal   Plane  Area.  —  Let 

F  =  area  of  a  horizontal  plane  surface ; 
z  =its  depth  below  free  surface  of  liquid: 
p  =  intensity  of  pressure  at  any  point; 
P=  resultant  pressure  on  area  F. 


Then 


P=Fp=wzF. 


The  center  of  pressure  (or  point  of  application  of  P)  is  evi- 
dently coincident  with  the  centroid  of  the  area;   for  the  forces 


RESULTANT  PRESSURE  ON   PLANE  AREA.  13 

of  which  P  is  the  resultant  are  parallel,  and  are  proportional  to 
the  elementary  areas  on  which  they  act. 

17.  Magnitude  of  Resultant  Pressure  on  any  Plane  Area. — 

If  a  submerged  plane  area  is  not  horizontal,  the  magnitude  of 
the  resultant  pressure  may  be  computed  as  follows: 

Let  AB  and  A'B'  (Fig.  4)  represent  two  vertical  projections 
of  the  surface,  its  inclination  to  the  horizontal  being  0.    Let 
2=  depth  of  an  elementary  area  dF  below  free  surface  of 

liquid; 
P=  resultant  pressure  on  whole  area  F. 


B 

FIG.  4. 


Then  the  total  pressure  on  the  element  dF  is 


hence  P  =  fpdF  =  wfz  dF, 

the  integration  being  extended  over  the  whole  area  F. 

If  z  is  the  value  of  z  at  the  centroid  of  the  area  F,  we  have 

f  z  dF=zF;    therefore     P  =  wzF. 

18.  Center  of  Pressure  of  Plane  Area.  —  To  determine  the 
point  of  application  of  the  resultant  pressure  P,  the  principle 
of  moments  must  be  employed. 

Referring  to  Fig.  4,  let  the  line  of  intersection  of  the 
plane  AB  with  the  free  surface  be  taken  as  axis  of  moments, 
and  let  the  moment  of  the  resultant  pressure  be  equated  to 


14  HYDROSTATICS. 

the  sum  of  the  moments  of  the  pressures  on  all  elementary 
areas. 

Let  the  distance  of  an  elementary  area  dF  from  the  axis  of 
moments  be  denoted  by  y,  the  corresponding  vertical  distance 
being  z,  so  that  z  =  y  sin  6.  Let  z,  y  refer  to  the  centroid  of  the 
area  F,  and  z'  ',  y'  to  the  center  of  pressure. 

The  equation  of  moments  is 


the  integration  covering  the  whole  area  F. 

Since  dP  =  wz  dF  =wy  sin  0-dF,  and  P  =  wzF=wj  sin  0-.F, 
the  equation  may  be  written 

wF  yyf  sm6  =  w  sin  0  •  f  y2  dF\ 
from  which 


The  numerator  of  this  value  is  equal  to  the  moment  of  iner- 
tia of  the  area  with  respect  to  the  line  in  which  its  plane  inter- 
sects the  free  surface  of  the  water;  the  denominator  is  the 
statical  moment  of  the  area  with  respect  to  the  same  axis. 
Denoting  these  quantities  by  /  and  G  respectively,  we  have 


If  the  area  has  an  axis  of  symmetry  which  is  perpendicular 
to  the  line  in  which  its  plane  intersects  the  water  surface,  the 
center  of  pressure  lies  in  this  axis  and  is  completely  determined 
by  equation  (1).  If  this  is  not  the  case,  a  second  moment- 
equation  is  required  for  the  complete  location  of  the  center  of 
pressure. 

Taking  as  axis  of  moments  a  line  lying  in  the  area  and 
perpendicular  to  the  axis  from  which  y  is  measured,  let  x  denote 


CENTER  OF  PRESSURE  OF  PLANE  AREA.  15 

the  distance  from  this  axis  of  an  element  dF;  then  the  equation 
of  moments  is 


Px'=      xP, 
which  may  be  written 

wF  yxr  sin  0  =  w  sin  0  •  j  xy  dF\ 
from  which 

jJ^J-  (2) 

yF     ~G  ........     (2) 

Here  J  denotes  the  product  of  inertia  of  the  area  F  with  respect 
to  the  axes  from  which  x  and  y  are  measured. 

EXAMPLES. 

Find  the  .magnitude  and  point  of  application  of  the  resultant  pressure 
on  the  area  described  in  each  of  the  following  examples,  the  liquid  being 
water. 

1.  A  rectangle  of  sides  4  ft.  and  6  ft.,  placed  vertically,  (a)  with 
shorter  side  in  water  surface,  (b)  with  longer  side  in  water  surface. 

Ans.  (a)  P  =  4500  Ibs.;  t/'  =  4  ft. 

2.  A  rectangle  with  sides  b,  d,  placed  vertically^  with  the  side  b  in 
the  water  surface.  Ans.  P  =  wbd*/2;  y'  =  %d. 

3.  A  circle  of  diameter  d}  its  plane  being  inclined  at  angle  0  to  the 
vertical,  and  the  center  being  distant  a  vertically  below  the  surface. 

Ans.  P  =  wnd*a/4i',  y'  =  asec6+d2/16asecd. 

4.  A  circle  of  2  ft.  diameter,  the  highest  point  being  6  inches  below 
the  surface,  and  the  plane  inclined  60°  to  the  vertical. 

Ans.  P  =  196  Ibs.;  y'  =  1\  ft. 

5.  A  semicircle  with  plane  vertical  and  diameter  in  the  surface. 

6.  A  triangle  of  base  2  ft.  and  altitude  3  ft.,  the  plane  being  vertical, 
(a)  with  vertex  in  water  surface  and  base  horizontal,  (6)  with  base  in 
surface.  Ans.  (a)  y'  =  ld.     (b)  y'  =  \d. 

7.  An  area  F,  whose  radius  of  gyration  about  a  horizontal  central 
axis  is  k,  placed  vertically,  with  centroid  at  depth  a  below  water  surface. 

Ans.  P=wFa-,  y'  = 


16  HYDROSTATICS. 

8.  The  vertical  plane  area  shown  in  Fig.  5. 


0 


5' 
5' 

15' 
FIG.  5. 


FIG.  6. 


9.  The  vertical  plane  area  shown  in  Fig.  6. 

Ans.  P  =  14,000  Ibs.;  x'  =  3.7l  ft., 


5.71  ft. 


19.  Pressure  Resolved  in  Given  Direction.  —  The  component, 
in  any  direction,  of  the  resultant  pressure  on  a  plane  area  may 
be  found  by  the  following  rule  : 

Pass  a  plane  through  the  centroid  of  the  area  perpendicular 
to  the  given  direction,  and  project  the  area  orthographically 
upon  it;  the  pressure  on  this  projected  area  is  equal  to  the 
required  component  of  the  resultant  pressure  on  the  given 
area. 

Thus,  let  it  be  required  to  find  the  component,  in  the  direc- 
tion MN,  of  the  resultant  pressure  on  the  area  AB  (Fig.  7). 


FIG.  7. 

The  value  of  the  resultant  pressure  is  P  =  wFz,  z  being  the 
depth  below  the  free  surface  of  C,  the  centroid  of  the  area  F. 
Through  C  pass  a  plane  perpendicular  to  MN,  and  let  A'Ef 
be  the  projection  of  AB  on  this  plane.  The  projected  area  is 
F'=Fcos6,  where  6  is  the  angle  between  P  and  MN]  the 


HORIZONTAL  PRESSURE  ON  CURVED  SURFACE.  17 

centroid  of  the  projected  area  coincides  with  that  of  the  given 
area  AB.     Resolving  P  in  the  direction  MN,  we  have 

P  cos  6  =  wF  z  cos  0, 
while  the  resultant  pressure  on  A'E'  is 

wF'z=wFz  cosO. 
These  values  being  equal,  the  proposition  is  proved. 

EXAMPLES. 

1.  Compute  the  horizontal  and  vertical  components  of  the  resultant 
pressure  on  a  rectangular  area  6  ft.  by  8  ft.,  inclined  30°  to  the  vertical, 
one  of  the  longer  edges  being  in  the  water  surface. 

Ans.  Hor.  comp.=6750  Ibs.;  vert.  comp.=3900  Ibs. 

2.  Compute  the  horizontal  and  vertical  components  of  the  resultant 
pressure  on  a  circular  area  4  ft.  in  diameter,  inclined  20°  to  the  vertical, 
the  center  being  5  ft.  below  the  water  surface. 

Ans.  Hor.  comp.=3690  Ibs.;  vert.  comp.=1345  Ibs. 

20.  Horizontal  Pressure  on  Curved  Surface.— In  case  the 
surface  is  not  plane,  the  resolved  pressure  in  a  given  direction 
cannot  in  general  be  computed  by  the  rule  above  given  for  a 
plane  surface  (Art.  19).  The  rule  does  hold,  however,  if  the 
direction  of  resolution  is  horizontal. 

Thus,  consider  the  pressure  upon  the  surface  AB  of  the 
submerged  body  X  (Fig.  8).  This  pressure  is  identical  with 


FIG.  8. 


that  upon  the  surface  AB  of  the  liquid  which  would  replace 
X  if  removed. 


HYDROSTATICS. 

Let  A'B'  be  the  orthographic  projection  of  the  surface  AB 
upon  any  vertical  plane,  and  consider  the  body  of  fluid  ABB' 'A' '. 
The  horizontal  component  of  the  pressure  on  A  B  is  counter- 
balanced by  the  pressure  on  A'B',  and  must  therefore  have  the 
same  magnitude  and  line  of  action. 


EXAMPLES. 

1.  A  circular  cylinder  4  ft.  long  and  2  ft.  in  diameter  is  placed  with 
axis  horizontal,  and  is  filled  with  water  to  a  depth  of  18  inches.    Com- 
pute the  magnitude  and  line  of  action  of  the  horizontal  thrust  of  the 
water  in  a  direction  perpendicular  to  the  axis  of  the  cylinder. 

Ans.  281  Ibs.  acting  in  line  1  ft.  below  water  surface. 

2.  A  hemispherical  bowl  2  ft.  in  diameter  is  filled  with  water.     Deter- 
mine the  magnitude  and  line  of  action  of  the  resultant  thrust  of  the 
water  on  the  bowl  in  a  given  horizontal  direction. 

Ans.  41.7  Ibs.;  .59  ft.  below  surface. 

3.  A  cylindrical  barrel  2  ft.  in  diameter  is  filled  with  water  to  a  depth 
of  30  inches.     Determine  the  magnitude  and  line  of  action  of  the  result- 
ant horizontal  thrust  on  half  the  interior  surface 

Ans.  391  Ibs.;  20  inches  below  water  surface. 

21.  Vertical  Pressure  on  Curved  Surface.— Let  AB  (Fig.  9) 
be  a  portion  of  the  surface  of  a  submerged  body  X,  and  let  it 
be  required  to  compute  the  resultant  of  the  vertical  components 


A' 


of  the  pressures  exerted  by  the  water  upon  the  elements  of 
AB.  Let  A'B'  be  the  orthographic  projection  of  AB  upon 
the  plane  of  the  water  surface.  The  volume  of  water  ABB' A' 
is  in  equilibrium  under  the  action  of  its  weight  and  the  pres- 


PRESSURE  ON  MASONRY  DAM.  19 

sures  acting  on  its  bounding  surface.  The  surface  A 'Br  being 
assumed  free  from  pressure,*  the  resolution  of  these  forces 
vertically  shows  that  the  vertical  component  of  the  pressure 
on  AB  is  equal  to  the  weight  of  the  body  of  water  ABB' A', 
and  that  its  line  of  action  passes  through  the  center  of  gravity 
of  this  body. 

In  the  case  shown  in  Fig.  9,  the  resultant  vertical  pressure 
of  the  water  on  the  surface  AB  of  the  body  X  acts  down- 
ward; while  in  such  a  case  as  that  shown  in  Fig.  10  it  acts 
upward.  But  in  the  latter  case,  as  in  the  former,  the  vertical 
pressure  is  equal  in  magnitude  to  the  weight  of  a  body  of 
water  ABB' A' . 

22.  Resultant  Pressure  on  Curved  Surface. — The  resultant 
pressure  on  a  curved  surface  is  not,  in  the  most  general  case, 
a  single  force,  since  a  system  of  forces  in  three  dimensions  is 
not  generally  equivalent  to  any  single  force.     In  practical  prob- 
lems it  will  usually  suffice  to  determine  the  effective  pressures 
in  horizontal   and  vertical   directions,   by   the   methods  just 
explained.     If  it  is  required  to  carry  the  reduction  of  the 
system  farther,  this  may  be  done  by  methods  explained  in 
works  on  Statics. f     It  is  sometimes  evident  from  symmetry 
that  the  total  pressure  is  equivalent  to  a  single  force.    This  is 
true  in  the  following  case. 

23.  Pressure  on   Masonry  Dam. — In  discussing  the  stability 
of  a  dam  it  is  necessary  to  compute  the  resultant  pressure 
exerted  upon  it  for  a  given  length,  say  one  foot.     The  horizontal 
and  vertical  components  of  this  pressure  may  be  computed  by 
the  rules  above  given,  and  since  these  components  act  in  the 
same  vertical  plane,  they  have  a  single  resultant  which  may 
readily  be  determined. 

*  Atmospheric  pressure  is  usually  neglected,  because  in  practical  cases 
it  is  commonly  the  excess  of  pressure  above  that  due  to  the  atmosphere  that 
is  of  importance.  It  may,  however,  be  taken  account  of  in  the  present  case 
by  conceiving  the  column  of  water  ABB' A'  extended  to  a  height  p0/w 
above  the  actual  water  surface,  p0  being  atmospheric  pressure. 

f  Theoretical  Mechanics,  Chapter  X. 


20 


HYDROSTATICS. 


FIG.  11. 


In  Fig.  11,  the  total  horizontal  pressure  H  on  the  face  AB 
is  equivalent  to  the  resultant  pressure  upon  A'B,  the  projection 

of  AB  on  a  vertical  plane;  while 
the  vertical  pressure  V  is  equal  to 
the  weight  of  the  body  of  water 
A  A'B.  The  line  of  action  of  H 
passes  through  the  center  of  pres- 
sure of  A'B,  while  that  of  V  passes 
through  the  center  of  gravity  of 
A  A'B',  their  intersection  gives  a 
point  in  the  line  of  action  of  P,  the 
resultant  of  H  and  V. 
In  masonry  dams  as  actually  constructed,  the  profile  AB 
often  differs  but  little  from  a  vertical  straight  line,  and  the 
vertical  component  of  the  pressure  is  neglected  in  discussing 
the  stability  of  the  dam.  This  is  an  error  on  the  side  of  safety 
in  the  design. 

Upward  pressure  on  base  of  dam. — If  a  masonry  dam  rests 
wholly  or  partly  upon  a  bed  of  porous  material  such  as  sand 
or  gravel,  which  is  continuous  with  the  bed  of  the  reservoir  so 
that  water  freely  enters  and  saturates  it,  an  upward  pressure 
results  which  may  have  an  important  effect  upon  the  stability 
of  the  dam.  If  the  water  is  at  rest  throughout  the  porous  bed, 
the  upward  pressure  must  be  computed  as  due  to  the  column 
of  water  A'B  (Fig. -11).  If  the  water  flows  through  the  gravel, 
as  it  will  unless  intercepted  by  an  impervious  barrier,  the  pres- 
sure will  decrease  in  the  direction  of  flow  (Arts.  90-93). 

Even  if  the  dam  rests  upon  impervious  rock,  a  hori- 
zontal fissure  in  the  masonry  may  permit  the  entrance  of 
water,  thus  causing  an  upward  pressure  upon  the  masonry 
above. 

The  following  examples  refer  to  a  dam  of  the  cross-section 
shown  in  Fig.  11,  the  dimensions  being  as  given  below.  In  all 
cases  the  required  pressures  are  to  be  computed  for  one  linear 
foot  of  the  dam. 

Referring  to  the  figure,  let  y  denote  depth  below  water  sur- 
face, x  the  corresponding  horizontal  distance  from  A'B  to  AB, 


CURVED  SURFACE   UNDER  UNIFORM   PRESSURE. 


21 


and  xr  the  horizontal  thickness  of  the  masonry.     The  values 
given  in  the  table  are  in  feet. 


y 

X 

*' 

y 

X 

x' 

-10 

34.8 

17.5 

80 

29.8 

57.5 

0 

34.3 

19.2 

90 

28.5 

66.0 

10 

33.8 

21.7 

100 

26.6 

75.7 

20 

33.4 

24.6 

110 

24.2 

86.5 

30 

32.9 

28.5 

120 

21.2 

98.2 

40 

32.4 

33.0 

130 

17.6 

111.0 

50 

31.9 

38.2 

140 

12.9 

125.3 

60 

31.5 

44.0 

150 

7.2 

140.7 

70 

30.6 

50.4 

160 

0.0 

159.5 

EXAMPLES. 

1.  Determine  the  magnitude,   direction,  and  line  of  action  of  the 
resultant  pressure  on  the  surface  AB. 

2.  Assuming  the  dam  to  rest  upon  saturated  gravel  for  its  entire 
thickness  BD,  compute  the  total  upward  force  if  the  pressure  is  every- 
where due  to  the  head  A'B. 

3.  Compute  the  total  upward  force  if  the  pressure  head  varies  uni- 
formly from  0  at  D  to  A'B  at  B. 

4.  If  the  material  of  the  dam  weighs  150  Ibs.  per  cu.  ft.,  compute 
the  magnitude  and  line  of  action  of  the  weight  per  linear  foot.     Deter- 
mine the  magnitude,  direction,  and  line  of  action  of  the  resultant  of 
the  water  pressure  and  the  weight  of  the  dam;  also  the  moment  of  this 
resultant  about  D.     (Solve  on  each  of  the  above  assumptions  as  to  the 
pressure  on  the  base.) 

24.  Curved  Surface  under  Pressure  of  Uniform  Intensity. — 
If  a  body  is  submerged  to  a  depth  which  is  great  in  comparison 
with  the  vertical  dimension  of  the  body,  the  variation  of  the 
intensity  of  pressure  on  its  bounding  surface  will  be  small  in 
comparison  with  its  actual  value,  and  for  practical  purposes 
this  variation  may  usually  be  disregarded. 

If  a  surface  is  under  pressure  of  uniform  intensity,  the  total 
resolved  pressure  in  any  direction  is  equal  to  the  resultant 
pressure  on  the  orthographic  projection  of  the  surface  upon  a 
plane  perpendicular  to  that  direction. 


22  HYDROSTATICS. 

Thus,  consider  the  pressure  on  the  part  AB  of  the  surface 
of  the  body  X  (Fig.  12).  If  it  is  required  to  compute  the 
resolved  part  of  this  pressure  in  any 
given  direction,  let  A'E'  be  the  ortho- 
graphic projection  of  AB  upon  a  plane 
perpendicular  to  that  direction.  If  X 
were  removed,  the  body  of  water  ABB'A' 
would  be  in  equilibrium  under  the  action 
of  the  pressures  exerted  by  the  surround- 

Fra  12  *n^  ^U^  ^^e  we*gkt  °f  this  k°dy  Demg 

by  supposition  negligible  in  comparison 

with  these  pressures).     Resolving  perpendicularly  to  A'B1 ',  the 

resolved    pressure   on  AB  exactly  balances  the   pressure  on 

A'&. 

EXAMPLES. 

1.  A  6-inch  pipe  carries  water  under  a  head  of  200  ft.     Compute 
the  total  pressure  per  foot  of  length  on  one  half  the  interior  surface. 

Ans.  6250  Ibs. 

2.  Compute  the  resultant  pressure  on  a  hemispherical  surface  2  ft.  in 
diameter  subjected  to  a  pressure  of  5  atmospheres.        Ans.  33,300  Ibs. 

25.  Resultant  Pressure  on  Submerged  Body. — The  resultant 
pressure  exerted  by  a  fluid  upon  a  body  which  is  wholly  or 
partly  submerged  in  it  is  a  force  equal  and  opposite  to  the 
weight  of  the  displaced  fluid,  and  its  line  of  action  passes 
through  the  center  of  gravity  of  the  displaced  fluid. 

It  is  obvious  that,  if  the  body  were  removed,  the  body  of 
fluid  which  would  replace  it  would  be  subjected  to  exactly  the 
same  pressure  as  that  actually  exerted  upon  the  submerged 
body.  But  the  body  of  fluid  would  be  in  equilibrium  under 
the  action  of  two  sets  of  forces:  (a)  the  weight  of  every  particle, 
with  resultant  acting  through  the  center  of  gravity;  (6)  the 
pressure  of  the  surrounding  fluid  upon  every  element  of  the 
bounding  surface.  The  resultants  of  these  two  sets  of  forces 
must  be  equal  in  magnitude,  opposite  in  direction,  and  col- 
linear. 

The  resultant  pressure  of  a  liquid  upon  a  body  wholly  or 


CONDITIONS  OF  EQUILIBRIUM  OF  A  FLOATING  BODY.      23 

partly  submerged  in  it  is  called  the  " buoyant  force."  The  cen- 
troid  of  the  submerged  volume  (the  point  of  application  of  the 
buoyant  force)  is  the  "  center  of  buoyancy." 

If  a  body  is  submerged  partly  in  one  fluid  and  partly  in 
another,  the  resultant  pressure  exerted  upon  it  by  both  is  equal 
to  the  total  weight  of  both  fluids  displaced.  Thus,  a  body  float- 
ing at  the  surface  of  water  displaces  a  certain  body  of  water 
and  a  certain  portion  of  air.  The  weight  of  the  displaced  air 
is  so  small  in  most  ordinary  problems  that  it  is  often  disre- 
garded. In  the  following  discussions  relating  to  floating  bodies 
the  pressure  of  the  air  will  not  be  considered  unless  specifically 
mentioned. 

26.  General  Conditions  of  Equilibrium  of  a  Floating  Body.— 

If  the  only  forces  acting  upon  a  floating  body  are  its  weight 
and  the  pressure  of  the  liquid,  these  two  forces  (or  strictly  sets 
of  forces)  must  balance  each  other.  This  requires 

(1)  That  the  weight  of  the  displaced  water  shall  equal  the 
weight  of  the  body,  and 

(2)  That  the  center  of  gravity  of  the  body  and  that  of  the 
displaced  water  shall  lie  in  the  same  vertical  line. 

From  these  two  conditions  it  is  possible  to  determine  what 
volume  of  water  will  be  displaced  by  a  body  of  known  weight; 
and  in  the  case  of  bodies  of  regular  shape  to  determine  by 
inspection  some  or  all  of  the  possible  positions  of  equilibrium. 


EXAMPLES. 

1.  A  plank  2"  by  12"  by  16',  weighing  40  Ibs.  per  cu.  ft.,  can  carry 
what  weight  without  sinking?  Ans.  60  Ibs. 

2.  A  box  closed  on  all  sides  is  made  of  lumber  1"  thick  weighing 
45  Ibs.  per  cu.  ft.     Its  outside  dimensions  are  2'  by  3'  by  6'.     If  half 
filled  with  water,  at  what  depth  will  it  float?  ^ 

3.  A  cone  of  specific  gravity  0.5  floats  with  axis  vertical  and  apex 
downward.     The  altitude  being  h  and  radius  of  base  a,  compute  the 
depth  of  flotation. 

4.  Compute  the  depth  of  flotation  of  the  cone  described  in  Ex.  3 
if  floating  with  vertex  upward. 


24  HYDROSTATICS. 

27.  Floating  Body  Acted  Upon  by  Any  Forces. — If  a  float- 
ing body  is  in  equilibrium  under  the  action  of  forces  additional 
to  the  pressure  of  the  liquid  and  the  weig^f  ^ie  body,  such 
additional  forces  must  be  included  in 
of  equilibrium. 

Since  the  weight  of  the  body  and  the  force  are 

both  vertical,  it  is  evident  that,  for  equilibrium,  the  additional 
forces  must  be  equivalent  either  to  a  vertical  fS^^pr  to  a 
couple. 


EXAMPLES. 


2'  b^§> 


1.  A  homogearneous  pallelopiped  1'  by  2'  by  3^p>f  specific  gravity 
0.25.  is  half  submerged  in  water,  one  face  being  horizontal.     What  force, 
besides  its  weight  and  the  pressure  of  the  water,  must  be^J|y^to  hold 
it  in  equilibrium? 

2.  If  the  same  body  is  in  equilibrium  with  a  diagonal  pi 
plane  of  the  water  surface,  determine  the  magnitude,  directiol 

line  of  action  of  the  resultant  force  acting  upon  the  body  in  addition  to* 
its  weight  and  the  buoyant  force. 

3.  A  right  prism  whose  bases  are  equilateral  triangles  floats  in  such 
a  position  that  a  median  plane  coincides  with  the  plane  of  the  water 
surface.     If  the  specific  gravity  is  0.8,  what  force,  besides  gravity  and 
the  buoyant  force,  must  be  acting  upon  the  body? 

28.  Stability  of  Equilibrium. — The  equilibrium  of  a  floating 
body  is  stable,  unstable,  or  neutral,  according  as  the  body  tends, 
after  being  slightly  displaced,  to  return  to  the  original  position, 
to  depart  farther  from  it,  or  to  remain  in  the  new  position. 

The  brief  discussion  of  stability  which  follows  is  limited  to 
the  case  in  which  the  only  forces  acting  on  the  body  are  its 
weight  and  the  pressure  of  the  liquid. 

A  complete  discussion  of  stability  would  require  a  consid- 
eration of  all  possible  displacements.  These  displacements  may 
be  resolved  into  translations  and  rotations. 

For  translations  the  nature  of  the  equilibrium  is  obvious, 
being  stable  for  vertical  displacements  and  neutral  for  horizon- 
tal displacements.  For  rotations  the  nature  of  the  equilibrium 
cannot  be  determined  so  simply. 


METACENTER.  25 

29.  Metacenter. — Let  a  floating  body  be  displaced  in  such 
a  way  that  the  submerged  volume  remains  constant.  Let  the 
body  in  the  new  position  be  represented  by  KMN  (Fig.  13), 


K 


LN  being  the  water  surface;  and  let  UN'  be  that  plane  in  the 
body  which  coincided  with  the  water  surface  in  the  original 
position  of  equilibrium. 

Let  Af  be  the  centroid  of  the  volume  L'M'N'  (being  there- 
fore the  position,  in  the  body,  of  the  center  of  buoyancy  in  the 
position  of  equilibrium),  and  A  the  centroid  of  the  volume 
LMN  (being  therefore  the  center  of  buoyancy  in  the  displaced 
position) .  Let  E'  be  the  center  of  gravity  of  the  floating  body. 

In  the  original  position  the  line  A'Bf  is  vertical.  In  the 
new  position  let  a  vertical  line  through  A  intersect  A'E'  (pro- 
duced if  necessary)  in  C.  If  the  angle  of  displacement  be  taken 
smaller  and  smaller  with  zero  as  limit,  C  approaches  a  definite 
limiting  position.  This  limiting  position  is  called  the  "meta- 
center." 

The  stability  of  the  equilibrium  can  be  tested  by  determin- 
ing the  position  of  the  metacenter. 

It  is  seen  that  in  the  displaced  position  the  forces  acting 
upon  the  body  are  equivalent  to  the  following  couple:  the 
weight  of  the  body  acting  downward  through  B',  and  the  buoy- 
ant force  acting  upward  through  A.  If  the  metacenter  C  falls 
above  B',  the  couple  tends  to  bring  the  body  back  to  the  original 
position  of  equilibrium;  if  C  falls  below  B' ,  the  couple  tends 
to  displace  the  body  still  farther.  Hence  in  the  former  case 
the  equilibrium  is  stable  and  in  the  latter  unstable;  while  if 
the  metacenter  coincides  with  the  center  of  gravity  of  the  body 
the  equilibrium  is  neutral. 


26 


HYDROSTATICS. 


< 

—  pi—  *~-  1 
-1*rJ 

f,         ^B 

j 

T 

fi 

j    j 

A  simple  rule  may  be  deduced  for  determining  the  position 
of  the  metacenter,  and  thus  testing  the  stability  of  the  equilib- 
rium. In  studying  the  stability  of  ships,  however,  it  is  not 
enough  to  test  whether  the  equilibrium  is  stable  for  small  dis- 
placements, but  the  degree  of  stability  for  both  small  and  large 
displacements  must  be  determined.  The  moment  of  the  couple 
consisting  of  the  weight  of  the  body  and  the  buoyant  force,  for 
any  displacement,  is  a  measure  of  the  degree  of  stability.  For 
a  vessel  of  known  shape  and  weight,  the  value  of  this  moment 

may  be  computed  for  any  angular 
displacement. 

30.  Variation  of  Pressure  in 
Rotating  Liquid.  —  If  a  body  of 
water  is  forced  to  maintain  a  con- 
dition of  uniform  rotation  about 
a  vertical  axis,  the  pressure  in- 
creases with  the  distance  from  the 
axis  of  rotation  in  a  manner  which 
can  be  estimated  as  follows  : 

Let  Fig.  14  represent  horizontal 
and  vertical  sections  of  a  cylin- 
drical vessel  filled  with  water.  If 
the  vessel  is  caused  to  rotate  uni- 
formly about  its  axis  of  figure,  the 
water  will  soon  take  up  a  rota- 
tional motion,  and  the  vessel  and  the  water  will  rotate  together 
as  if  forming  one  rigid  body. 

In  order  to  determine  the  variation  of  pressure  with  the  dis- 
tance from  the  axis  of  rotation,  consider  a  prismatic  element  of 
water  of  uniform  cross-section  F  whose  axis  is  horizontal  and  co- 
planar  with  the  axis  of  rotation.    Such  an  element  is  represented 
in  vertical  projection  at  AB  and  in  horizontal  projection  at  A'B'  '. 
Let  7*1  =  distance  of  (A,  A')  from  axis  of  rotation; 
r2=       "        "  (£,•£')     "       "    " 
pi  =  in  tensity  of  pressure  at  point  (A,  A'); 


FIG.  14. 


co  =  angular  velocity  of  rotation. 


VARIATION   OF  PRESSURE  IN  ROTATING  LIQUID.          27 

The  fundamental  equation  of  dynamics, 
force  =  mass  X  acceleration, 

applies  to  the  motion  of  any  body  whatever,*  force  meaning  the 
resultant  of  all  external  forces  acting  on  the  body,  mass  the 
total  mass  of  the  body,  acceleration  the  acceleration  of  its  mass- 
center.  Applying  this  to  the  elementary  prism  AB,  it  is  seen 
that  the  acceleration  of  the  mass-center  and  the  resultant  force 
have  the  direction  BA,  since  the  mass-center  describes  a  circle 
whose  center  lies  in  the  axis  of  rotation.  The  radius  of  this 
circle  being  J(ri  +r%)t  the  acceleration  has  the  value 


The  only  forces  not  perpendicular  to  AB  are  the  pressures  on 
the  ends  of  the  element,  and  the  resultant  of  these  is 


The  mass  of  the  element  is 


The  dynamical  equation  therefore  reduces  to  the  form 


w      w  2g 

The  variation  of  the  pressure  in  the  vertical  direction  follows 
the  same  law  as  if  the  body  of  water  were  at  rest.  Thus,  con- 
sider a  prismatic  element  CD  (Fig.  14),  whose  axis  is  vertical. 
If  force  and  acceleration  be  resolved  vertically,  it  is  seen  that, 
since  the  acceleration  in  the  vertical  direction  is  0,  the  sum  of 
the  vertical  components  of  all  forces  acting  upon  the  element 
must  be  0.  The  forces  having  vertical  components  are  the  nor- 
mal pressures  on  the  ends  of  the  prism  and  the  weight  of  every 
particle  of  the  water.  If  z\  and  z2  are  the  heights  of  D  and  C 
respectively  above  any  horizontal  plane,  pi  and  p2  the  corre- 
sponding values  of  the  pressure-intensity,  and  F  the  cross-sec- 
tion of  the  element,  we  have  for  the  total  downward  force 


*  Theoretical  Mechanics,  Art.  380. 


28 


HYDROSTATICS. 


Equating  this  to  0, 


P2        Pi 

=Z\  — 

W         W 


(2) 


The  results  expressed  by  equations  (1)  and  (2)  may  be  com- 
bined into  a  single  equation  as  follows : 

Let  A  and  B  be  any  two  points  in  the  rotating  liquid;  zi, 
Z2  their  ordinates  from  some  horizontal  plane;  r\,  r2  their  dis- 
tances from  the  axis  of  rotation;  pi}  p2  the  pressures  at  the 
two  points  respectively.  Then 


p2 

- 


or,  in  symmetrical  form, 


pi 
w 


w 


(3) 


.    ..     .     (4) 


It  is  not  difficult  to  show  that  this  equation  holds  even  if 
the  axis  of  rotation  is  not  vertical.  In  such  a  case  the  value 
of  z  for  a  given  particle  will  continually  change,  since  it  must 
be  measured  from  a  horizontal  plane,  irrespective  of  the  direc- 
tion of  the  axis  of  rotation.  The  form  of  the  containing  vessel 
is  obviously  of  no  consequence. 


31.  Form  of  Free  Surface  of  Rotating  Liquid. — It  may  be 

shown  that,  if  a  body  of  liquid  rotates  uniformly  about  a  ver- 
tical axis,  the  upper  surface,  if 
free,  will  assume  the  form  of  a 
paraboloid  of  revolution. 

Let  A  (Fig.  15)  be  the  point 
in  which  the  axis  of  rotation  pierces 
-  the  free   surface,  and  B  another 
point  of  the  free  surface.     Let  z 
denote  the  height  of  B  above  a 
FlG>  15-  horizontal  plane  through  A,  and  r 

the  distance  of  B  from  the  axis  of  rotation.     Then  in  equation 


A 


r — ; 


FORM  OF  FREE  SURFACE  OF  ROTATING  LIQUID.          29 

(4)  we  may  put  z\  =  0,  22  =  z,  p\  =p2  =  0,  r*i  =0,  r^  =r;  and  the 
equation  becomes 


(5) 


Evidently  r,  2  are  the  coordinates  of  the  curve  cut  from  the 
free  surface  by  a  plane  containing  the  axis  of  rotation.  Equa- 
tion (5)  represents  a  parabola  with  vertex  at  A  and  principal 
diameter  vertical. 

Obviously  the  same  equation  results  if  the  pressures  at  A 
and  B  have  any  equal  values;  all  surfaces  of  equal  pressure 
are  therefore  alike. 

EXAMPLES. 

1.  A  body  of  water  rotates  uniformly  about  a  vertical  axis,  making 
80  revolutions  per  minute.     If  the  upper  surface  is  free,  what  is  the 
difference  in  level  between  its  lowest  point  and  a  point  16  inches  from 
the  axis?  Ans.  1.95  ft. 

2.  Does  the  variation  of  pressure  due  to  rotation  depend  upon  the 
density  of  the  liquid?    What  is  the  result  of  Ex.  1  if  the  liquid  is  mer- 
cury (specific  gravity  13.6)? 


CHAPTER  III. 

FLOW  OF  WATER  THROUGH  ORIFICES.       TORRICELLI'S 

THEOREM. 

32.  Stream   of   Water  with  Steady  Flow.— The  streams  of 
water  with  which  hydraulic  discussions  and  experiments  are 
concerned  may  be  either  confined  in  pipes,  partly  confined  in 
open  channels,  or  wholly  unconfined. 

If,  at  every  cross-section  of  a  stream,  the  velocity  of  flow 
and  the  form  and  size  of  the  cross-section  remain  constant  (the 
conditions  at  different  sections,  however,  not  necessarily  being 
alike),  the  flow  is  said  to  be  steady. 

The  most  important  practical  cases  are  of  the  kind  thus 
described,  and  to  such  the  discussion  will  for  the  most  part  be 
restricted. 

33.  Rate  of  Discharge. — By  rate  of  discharge  of  a  stream  is 
meant  the  quantity  of  water  passing  a  given  cross-section  per 
unit  time.     It  will  usually  be  expressed  in  cubic  feet  per  second. 

It  is  evident  that,  so  long  as  the  condition  of  flow  remains 
steady,  the  rate  of  discharge  has  the  same  value  at  all  cross- 
sections,  as  well  as  a  constant  value  at  any  given  section. 

34.  Velocity  in  a  Cross-section. — It  is  impossible  to  deter- 
mine, either  in  magnitude  or  in  direction,  the  velocities  of  all 
the  various  particles  which,  at  any  instant,  are  passing  a  given 
cross-section  of  a  stream.     Even  at  a  section  where  the  stream 
is  neither  converging  nor  diverging  (as  at  A,  Fig.  16),  the  direc- 
tions of  motion  of  the  different  particles  doubtless  differ,  at 
least  slightly,  from  the  direction  of  the  axis  of  the  stream, 

30 


MEAN   VELOCITY  IN  A  CROSS-SECTION.  31 

although  the  predominating  motion  has  that  direction.  Still 
more  important,  probably,  are  the  irregularities  in  the  mag- 
nitudes of  the  velocities;  especially  in  case  of  a  confined  stream, 
in  which  the  particles  adjacent  to  the  confining  surface  are 
retarded  by  friction. 

In  a  section  where  the  stream  converges  or  diverges  (as  at 
B  or  D)  the  variation  in  direction  of  the  velocities  of  particles 
in  different  parts  of  the  cross-section  is  doubtless  still  more 
important. 

The  component  of  velocity  in  the  direction  of  the  axis  of 
the  stream  is  the  only  component  usually  considered.  In  the 
following  discussions,  therefore,  it  is  commonly  to  be  under- 
stood that  "velocity"  means  "axial  component  of  velocity." 


o 

FIG.  16. 

35.  Mean   Velocity  in  a  Cross-section. — Mean  velocity  in  a 
cross-section  may  be  defined  as  the  quotient  of  the  rate  of  dis- 
charge by  the  area  of  the  cross-section. 

Let  q  =rate  of  discharge  (cu.  ft.  per  sec.); 
F  =  area  of  cross-section  (sq.  ft.); 
v=mean  velocity  in  cross-section  (ft.  per  sec.); 

then  v  =  p>    or    q=Fv> 

36.  Values  of  Mean  Velocity  in  Different  Cross-sections. — 
In  case  of  steady  flow,  the  mean  velocity  remains  constant  at 
any  given  cross-section,  but  has  different  values  at  different 
sections  if  the  cross-sectional  areas  are  unequal.     If  the  values 
of  the  area  at  different  sections  are  denoted  by  F\,  F'%,  .  . . , 
and  the  corresponding  values  of  the  mean  velocity  by  vi,  v-2, 
. .  . ,  we  have 

FiVi  =^2^2=  •  •  •  =q=  constant. 

The  mean  velocity  is  thus  inversely  proportional  to  the  area  of 
cross-section. 

The  above  equation  is  called  the  equation  of  continuity. 


32  FLOW  OF  WATER  THROUGH  ORIFICES. 

37.  Torricelli's  Theorem.  —  If  a  small  orifice  be  opened  in  the 
side  of  a  vessel  containing  water,  the  velocity  of  the  escaping 
jet  will  be  nearly  equal  to  the  velocity  acquired  by  a  body 
falling  freely  from  rest  through  a  vertical  distance  equal  to  the 
depth  of  the  orifice  below  the  free  surface  of  the  water. 

Let  h  denote  the  depth  of  the  orifice  below  the  free  surface, 
and  v  the  velocity  of  the  jet;  then  the  proposition  states  that 
the  following  equation  is  nearly  satisfied  : 


the  actual  value  of  v  being  a  little  less  than  that  given  by  the 
equation. 

The  truth  of  the  proposition  is  known  from  experiment,  and 
it  is  inferred  that,  if  frictional  resistances  could  be  eliminated, 
the  equation  v2  =  2gh  would  be  exactly  satisfied.  It  will  be 
shown  later  that  this  conclusion  follows  from  the  principle  of 
energy. 

The  depth  of  the  center  of  a  small  orifice  below  the  free  sur- 
face of  the  water  is  called  the  head  on  the  orifice. 

38.  Actual  Velocity  of  Jet.  —  The  actual  mean  velocity  of  the 
jet  is  always  less  than  that  computed  from  the  formula  v2  =  2gh't 
how  much  less  depends  upon  the  nature-  of  the  orifice. 

In  Fig.  17  are  represented  four  cases.  The  orifices  are  all 
supposed  to  be  circular,  the  smallest  diameters  being  equal,  and 

all  are  supposed  to  be  under  the 
same  head. 

In  case  of  the  sharp-edged 
orifice  shown  at  A,  the  stream 
converges  as  it  passes  the  plane 
of  the  orifice.  The  smallest 
cross-section  of  the  jet  is  found 
at  S,  and  at  this  section  the 
mean  axial  velocity  has  its 
FlG-  17.  greatest  value.  For  such  an 

orifice  the  value  of  the  mean  velocity  is  but  slightly  less  than 
the  value  computed  from  the  above  formula.  This  is  because 


ACTUAL  VELOCITY  OF  JET.  33 

the  frictional  resistances  (which  depend  upon  the  velocities  with 
which  adjacent  portions  of  fluid  slide  over  each  other  or  over 
other  bodies)  are  in  this  case  comparatively  small.  Within  the 
vessel  the  velocities  of  the  particles  are  small,  and  the  only 
place  where  the  friction  becomes  important  is  near  the  orifice. 
The  surface  of  contact  between  the  jet  and  the  vessel  is  so 
small  that  the  retarding  effect  is  slight. 

A  cylindrical  orifice  in  a  plate,  such  as  is  shown  at  B,  Fig. 
17,  gives  practically  the  same  result  as  the  sharp-edged  orifice 
at  A,  if  the  thickness  of  the  plate  is  small.  If,  however,  the 
thickness  is  increased,  the  condition  of  the  jet  changes  mate- 
rially. 

Thus,  consider  a  cylindrical  orifice  in  a  thick  plate,  as  shown 
at  (7,  Fig.  17.  As  the  jet  enters  the  orifice  it  tends  to  converge 
(as  in  preceding  cases) .  The  air  surrounding  the  jet  at  the  con- 
tracted portion  within  the  orifice  is,  however,  quickly  carried 
out  by  friction,  thus  reducing  the  pressure  below  that  of  the 
atmosphere;  this  causes  the  stream  to  expand  and  fill  the  ori- 
fice. Whether  this  result  is  complete  or  partial  will  depend 
upon  the  thickness  of  the  plate.  The  frictional  resistances 
between  the  particles  of  water  (due  to  the  irregular  motions 
set  up  within  the  orifice),  and  also  the  friction  against  the 
cylindrical  surface  of  the  orifice,  are  considerably  greater  in 
this  case  than  in  the  preceding,  and  the  velocity  of  the  jet  is 
correspondingly  less. 

In  case  of  an  orifice  with  rounded  inner  edge  (D,  Fig.  17), 
the  frictional  resistance  due  to  the  surface  of  the  orifice  is 
greater  than  in  the  case  of  the  sharp-edged  orifice,  because 
the  area  of  contact  is  greater.  The  internal  friction  due 
to  irregularities  of  motion  of  the  particles  is,  however, 
less  than  in  case  C.  The  velocity  of  the  jet  in  the  case 
shown  at  D  is  greater  than  in  case  C,  but  less  than  in  cases  A 
and  B. 

In  all  cases,  the  " velocity  of  the  jet"  is  taken  to  mean  the 
average  velocity  in  a  section  outside  the  orifice  at  a  point  where 
the  jet  has  become  cylindrical  (sections  such  as  are  marked  S 
in  the  four  cases  shown  in  Fig.  17). 


34  FLOW  OF  WATER   THROUGH  ORIFICES. 

39.  Rate  of  Discharge  from  Small  Orifice. — The  rate  of  dis- 
charge from  an  orifice  is  equal  to  the  product  of  any  cross- 
sectional  area  of  the  stream  into  the  mean  axial  velocity  in  that 
cross-section.    Although  this  is  true  for  any  section,  it  is  com- 
mon to  consider  the  section  at  which  the  stream  has  become 
cylindrical.' 

If  the  four  cases  above  discussed  be  compared  with  refer- 
ence to  the  rate  of  discharge  (the  smallest  cross-section  of  the 
orifice  having  the  same  value  in  all  cases),  it  will  be  seen  that 
the  greatest  discharge  does  not  necessarily  accompany  the 
greatest  velocity.  Thus,  experiment  shows  that  case  D  gives 
the  greatest  discharge  and  case  C  the  next  greatest.  The  reason 
is  that  the  greater  size  of  the  jet  at  S  in  these  cases  more  than 
counterbalances  the  greater  velocity  found  in  cases  A  and  B. 

40.  Coefficient^?  Velocity. — The  factor  which,  applied  to  the 
ideal  velocity  \/2gh,  gives  the  true  mean  velocity  of  the  jet  is 
called  the  coefficient  of  velocity.    This  coefficient  is  an  abstract 
number  less  than  unity. 

41.  Coefficient  of  Contraction. — The  ratio  of  the  area  of  the 
cross-section  of  the  jet  to  that  of  the  orifice  is  called  the  co- 
efficient  of   contraction.     This    is    also    an    abstract    number, 
and   in  most  cases  of  practical  importance  its  value  is  less 
than   unity. 

42.  Ideal  Velocity  and   Discharge. — Practical  formulas  for 
velocity  and  discharge  from  small  orifices  are  obtained  by 
applying  experimental   coefficients   to  formulas  which  would 
hold  in  a  certain  ideal  case.      This  ideal  case  is  one  in  which 
there  are  supposed  to  be  no  frictional  resistances  to  affect  the 
velocity,  and  in  which  the  cross-section  of  the  jet  is  supposed 
to  be  equal  to  that  of  the  orifice. 

If  F  denotes  the  area  of  the  orifice  and  h  the  head  on  its 
center,  the  ideal  velocity  is  given  by  the  formula 


ACTUAL  VELOCITY  AND  DISCHARGE.  35 

and  the  ideal  discharge  (per  unit  time)  by  the  formula 


43.  Actual  Velocity  and  Discharge.  —  If  c'  denotes  the  coeffi- 
cient of  velocity  and  c"  that  of  contraction,  the  true  values  of 
the  mean  velocity  of  the  jet  and  the  rate  of  discharge  may  be 
written 

v=cf\/2gh; 
q=c"Fv=c'c"F\/2gh. 

44.  Coefficient  of  Discharge.  —  The  ratio  of  the  actual  value 
of  the  rate  of  discharge  to  its  ideal  value  is  called  the  coefficient 
of  discharge. 

If  the  value  of  this  coefficient  is  c,  we  have  from  the  defini- 
tion 


and  therefore 

c=c'c". 

45.  Circular  Standard  Orifice.  —  An  orifice  with  sharp  edge 
(as  A,  Fig.  17),  or  an  orifice  in  a  thin  plate  (as  B,  Fig.  17),  is 
called  a  standard  orifice.     The  values  of  the  coefficients  of  veloc- 
ity and  contraction  for  small  circular  orifices  of  this  kind  have 
been  fairly  well  established  by  experiment.     The  following  may 
be  taken  as  sufficiently  near  the  true  values  : 

c'=0.98;    c"=0.62;    c=cV'=0.61. 

46.  Discharge  from  Large  Orifice.  —  If  an  orifice  is  not  small 
in  comparison  with  the  head  on  its  center,  it  may  be  necessary, 
in  estimating  the  discharge,  to  take  account  of  the  different 
values  of  the  head  for  different  parts  of  the  cross-section.     For 
the  ideal  case  described  in  Art.  42  the  value  of  the  discharge 
per  unit  time  may  be  found  as  follows: 


Let  dF  =  area  of  a  differential  element  of  the  cross-section; 
z  =head  on  element  dF; 

then  the  discharge  per  unit  time  through  this  element  is  V2gz  •  dF, 


36  FLOW  OF  WATER  THROUGH  ORIFICES. 

and  the  discharge  per  unit  time  for  the  whole  orifice  is  the 
integral  of  this  expression  for  the  entire  area  F. 

For  the  actual  case,  the  rate  of  discharge  is  found  by  apply- 
ing a  coefficient  to  this  ideal  value.  The  result  may  be 
written 


q=cV2gfz*dF. 


47.  Large  Horizontal  Orifice. — If  the  plane  of  the  orifice  is 
horizontal,  z  is  constant  and  equal  to  h.    The  formula  therefore 
reduces  to 

q  =  cF\/2gh, 

identical  in  form  with  the  formula  for  the  case  of  a  small  orifice. 
The  value  of  c  is  found  to  vary  with  the  form  of  the  orifice; 
and  for  circular  orifices  it  doubtless  differs  somewhat  from  the 
value  applying  to  very  small  orifices. 

48.  Large  Vertical  Orifice. — In  order  to  apply  the  above 
general  formula  to  the  case  in  which  the  plane  of  the  orifice  is 
not  horizontal,  the  form  and  dimensions  of  the  orifice  must  be 
given.    The  most  important  case  practically  is  that  of  a  rect- 
angular orifice  with  one  pair  of  edges  horizontal. 

49.  Rectangular  Orifice. — Consider  a  rectangular  orifice  of 

horizontal  width  b  and  depth  d  (Fig. 
18). 

Let  hi  =  head  on  upper  edge ; 
h2  =head  on  lower  edge; 
h  =  head  on  center. 
To  apply  the  general  formula  of 
Art.  46,  let  the  differential  area  be 

an  elementary  strip  of  length  b  (horizontal)  and  width  dz  (ver- 
tical). Then 


q=cV2gb 

* 

If  the  head  on  the  center  is  great  in  comparison  with   the 


RECTANGULAR   NOTCH  OR  WEIR.  37 

vertical  dimension  of  the  orifice,  a  simpler  approximate  expres- 
sion may  be  used.    Thus  we  have 

,       j     d      -.       ,     d 
n2=fi-l--;    hi=h~-~ 

Expanding  (h  +d/2)f  and  (h-d/2)*  by  the  binomial  theorem  and 
substituting  in  the  above  formula,  the  result  becomes 


in  which  the  series  converges  rapidly  except  for  relatively  large 
values  of  d/h.    If  all  terms  except  the  first  be  neglected,  we  have 

q  =  d)d\'  2gh, 

which  is  identical  with  the  formula  for  discharge  through  a  small 
orifice. 

Whatever  the  form  of  the  orifice,  the  formula  obtained  by 
taking  account  of  the  actual  head  on  every  part  of  the  orifice 
differs  little  from  that  obtained  by  using  the  head  on  the  cen- 
troid  as  applying  to  all  parts  of  the  area,  if  h  exceeds  a  small 
multiple  of  the  vertical  dimension  of  the  orifice. 

50.  Rectangular  Notch  or  Weir.  —  In  practical  Hydraulics  the 
most  important  case  of  rectangular  orifice  is  that  in  which  the 
upper  side  is  open.  The  formula  for  this  case  is  obtained  by 
putting  hi  =  0.  Thus,  using  a  coefficient  of  discharge,  and  re- 
placing h2  by  H,  the  actual  value  of  q  may  be  written 


51.  Triangular  Notch.  —  Another  case  of  some  importance  is 
that  of  a  triangular  notch  (Fig.  19). 

Let  b  =  width  of  notch  at  water  surf  ace; 
H  =  head  on  vertex. 

Take  as  elementary  area  a  horizon- 
tal strip  at   depth  z,  of  vertical  width  FIG.  19. 

dz.    The  length  of  this  strip  is  b(H-z)/H,  and 


38  FLOW  OF  WATER  THROUGH  ORIFICES. 

The  value  of  the  rate  of  discharge  for  the  whole  area  is 


or 


c  being  the  coefficient  of  discharge. 

The  values  of  the  coefficient  of  discharge  for  this  case  and 
the  preceding  cases  will  be  considered  in  Chapter  XIII,  in  which 
the  use  of  orifices  and  weirs  for  the  measurement  of  rate  of  dis- 
charge is  discussed. 

52.  Unequal  Pressures  on  Water-surface  and  Jet. — In  the 

ordinary  experiments  upon  which  Torricelli's  theorem  (Art.  37) 
is  based,  the  surface  of  the  water  is  under  atmospheric  pressure, 
and  the  jet  discharges  into  the  atmosphere.  The  theorem  still 
holds  if  the  pressures  at  these  two  points  differ  from  that  of  the 
atmosphere,  so  long  as  the  two  are  equal.  If,  however,  unequal 
pressures  exist  at  the  water  surface  and  the  point  of  discharge, 
the  theorem  must  be  modified. 

Consider  first  the  case  in  which  the  pressure  at  the  water 
surface  exceeds  that  of  the  atmosphere  by  p1}  while  the  jet 
discharges  into  the  atmosphere.  Thus  the  space  above  the 
water  (at  A,  Fig.  20)  may  contain  compressed  air.  Evidently 
the  effect  of  the  pressure  pi  on  the  conditions  existing  within 
the  body  of  water  is  the  same  as  that  of  an  additional  depth  of 


FIG.  20. 


FIG.  21. 


water.  The  height  of  water  column  necessary  to  produce  a 
pressure  of  intensity  pi  is  pi/w;  hence  the  discharge  takes 
place  as  if  under  a  head  h+pi/w  with  equal  pressures  at  the 
water  surface  and  jet. 


SUBMERGED  ORIFICE. 


39 


Next  suppose  the  discharge  to  take  place  into  a  closed 
chamber  (Fig.  21)  in  which  the  pressure  exceeds  that  of 
the  atmosphere  by  p2.  The  effect  of  this  pressure  p2  is  the 
same  as  that  of  a  diminution  of  the  head  on  the  orifice  by  the 
amount  p2/w. 

It  appears,  therefore,  that  Torricelli's  theorem  may  be 
applied  to  the  case  of  unequal  pressures  if  the  head  on  the 
orifice  be  corrected  for  the  difference  of  the  pressures.  Thus, 
if  the  pressures  at  A  and  B  (Fig.  21)  are  pi  and  p2  respectively, 
the  value  of  the  ideal  velocity  is 


w 


EXAMPLES. 

1.  If  the  depth  of  the  orifice  below  the  water  surface  (Fig.  21)  is 
4  ft.,  the  pressure  at  A  atmospheric,  and  that  at  B  absolute  zero,  com- 
pute the  velocity  of  the  jet.     What  head  would  produce  the  same  velocity 
if  the  pressures  at  A  and  B  were  equal?  Ans.  v  =  49A  ft.  per  sec. 

2.  If  in  Fig.  21  h  =  40  ft.,  pi  =  absolute  zero,  p2  =  atmospheric  pressure, 
compute  the  velocity  of  the  jet. 

53.  Submerged  Orifice. — If  the  discharge  takes  place  as  in 
Fig.  22,  the  orifice  being  below  the  surface 
of  the  water  in  the  receiving  chamber,  the 
jet  at  B  is  under  a  pressure  wh2  in  addition 
to  the  pressure  existing  at  the  surface  D. 
The  head  on  the  orifice  must  therefore 
be  corrected  by  the  subtraction  of  h2, 
in  applying  Torricelli's  theorem.  That  is,  FIG.  22. 

the  ideal  velocity  is 


D 


v=\/2g(h1-h2). 

In  applying  this  formula  to  actual  cases,  the  coefficients  of 
velocity  and  discharge  must  be  given  different  values  from  those 
applying  to  similar  orifices  in  case  of  discharge  into  air. 

The  actual  values  of  coefficients  of  discharge  for  some  of 


40  FLOW  OF   WATER  THROUGH  ORIFICES. 

the  cases  of  most  practical  importance  will  be  given  in  Chapter 
XIII. 

The  foregoing  principles  will  receive  theoretical  justification 
in  the  following  chapters,  in  which  the  theory  of  energy  is  applied 
to  all  cases  of  steady  flow. 

54.  Discharge  under  Varying  Head. — If  the  head  on  an  orifice 
varies,  the  total  discharge  in  a  given  time,  or  the  time  required 
for  a  given  total  discharge,  can  be  computed  only  by  integra- 
tion. 

Let  the  head  on  the  orifice  at  any  instant  be  y,  the  rate  of 
discharge  at  that  instant  being  q,  and  let  c  be  the  coefficient  of 
discharge;  then 

q=cFV2gy. 

If  dQ  denotes  the  volume  discharged  in  the  time  dt, 
dQ=qdt=cF\/2gy-dt. 

In  a  particular  case  dQ  can  be  expressed  in  terms  of  y  and  dy 
and  the  equation  can  be  integrated. 

55.  Time  of  Emptying  a  Reservoir. — Consider  a  vessel  or 

reservoir  of  any  form,  filled  with  water 
to  a  certain  level,  and  let  it  be  required 
to  determine  the  total  time  required  for 
the  surface  to  fall  any  given  amount. 
It  will  be  assumed  that  the  coefficient 
of  discharge  does  not  vary  with  the 
head.  (See  Fig.  23.) 
23.  Let  y=  head  on  center  of  orifice  at 

time  t; 

2/i  =  initial  value  of  y; 
2/2  =  final  value  of  y; 

A  =  horizontal  cross-section  of  reservoir  at  water  surface 
(variable) ; 


TIME  OF  FILLING  A  RESERVOIR. 


41 


F  =  area  of  orifice ; 

Q  =  total  volume  discharged  from  some  assumed  instant 

up  to  the  instant  t\ 
q=  rate  of  discharge  at  time  t; 
c  =  coefficient  of  discharge,  assumed  constant. 
Then,  as  in  the  preceding  article, 


But  also 


dQ=qdt=cFV2gy-dt. 
dQ=-Ady 


(the  minus  sign  being  used  because  Q  increases  as  y  decreases) 
hence 

cFV2gy-dt  =  -Ady. 

If  A  is  variable,  it  must  be  known  as  a  function  of  y  in 
order  that  the  solution  may  be  completed. 

Reservoir  of  uniform  horizontal  cross-section. — Let  A  be  con- 
stant, and  let  T  denote  the  time  required  for  y  to  change  from 
?/i  to  y2.  Integrating  the  last  equation  between  the  stated 
limits, 


56.  Time  of  Filling  a  Reservoir. — If  water  flows  from  one 
reservoir  or  chamber  into  another  through  a  submerged  orifice, 
the  pressure  against  which  the  flow 
takes  place  increases  as  the  water 
surface  rises  in  the  receiving 
chamber.  If  the  horizontal  area 
of  the  supply  reservoir  is  great  in 
comparison  with  that  of  the  receiv- 
ing chamber,  the  drop  in  the  sur- 
face of  the  former  will  be  inap-  pIG  24 
preciable.  In  this  case  let  y  denote 

the  difference   in   level   between   the    water    surfaces  in  the 
two  reservoirs  at  any  instant,  the  remaining  notation  being  as 


42  FLOW  OF  WATER  THROUGH  ORIFICES. 

in  the  preceding  case.    Then  the  reasoning  of  Art.  55  applies 
without  change,  leading  to  the  same  formula. 

57.  Canal  Lock. — A  canal  lock  consists  of  a  chamber  or  com- 
partment which  is  placed  in  communication  alternately  with 
two  bodies  of  water  whose  surfaces  are  at  different  levels.  The 
time  of  emptying  or  filling  the  lock  may  be  estimated  by  the 
above  formula,  with  proper  value  of  c. 

If  the  discharge  into  or  from  the  lock  takes  place  through 
a  tunnel,  the  coefficient  of  discharge  will  be  much  less  than  for 
a  simple  orifice.  Its  value  will  depend  upon  the  length  of  the 
tunnel,  the  form  and  size  of  its  cross-section,  and  the  roughness 
of  its  surface.  The  problem  will  be  analogous  to  that  of  esti- 
mating the  effect  of  friction  on  the  flow  in  a  pipe,  to  be  discussed 
in  Chapter  IX. 

EXAMPLES. 

1.  A  reservoir   of  1000    sq.  ft.  horizontal  cross-section   is  emptied 
through  an  orifice  of  area  2  sq.  ft.     Taking  the  head  on  the  center  of 
the  orifice  as  initially  19  ft.,  compute  the  times  required  for  the  surface 
to  drop  1  ft.,  3.5  ft.,  and  7  ft.  respectively.    Take  c  =  0.65. 

Ans.  22.2  sec.;   81.0  sec.;  172  sec. 

2.  With  initial  conditions  as  in  Ex.  1,  compute  the  drop  of  the  water 
surface  in  3.5  min  and  in  7  min.  Ans.  8.3  ft.;  14.3  ft. 

3.  Two  reservoirs  of  horizontal  cross-sections  A',  A",  are  connected 
by  a  submerged  orifice.     If  the  difference  in  level  of  the  two  water  sur- 
faces at  any  instant  is  denoted  by  y,  show  that  the  time  required  for  y 
to  change  from  y\  to  y.2  is  given  by  the  formula 


in   which   A  =  A'A"/(A'+A").    This   includes   the  two    cases   above 
treated,  reducing  to  one  of  them  if  A'  or  A"  is  infinite. 

4.  As  a  particular  case  of  Ex.  3,  let  A  =  24  sq.  ft.,  4"  =  10  sq.  ft., 
F = 0.5  sq.  ft.,  yl  =  15  sq.  ft.,  yz  =  0,  c  =  0.6.     Determine  T. 


CHAPTER  IV. 
THEORY  OF  ENERGY  APPLIED  TO  STEADY  STREAM  MOTION.* 

58.  Transformation  and  Transference  of  Energy  in  Steadily 
Flowing  Stream. — Let  AB  (Fig.  25)  represent  a  portion  of  a 
steady  stream,  the  direction  of  flow  being  from  A  toward  B. 
Consider  the  transformations  and  transferences  of  energy  in 
which  any  given  particle  of  water  is  concerned. 

The  energy  possessed  by  a  particle  at  any  instant  is  in  gen- 
eral part  potential  and  part  kinetic. 

A  particle  of  mass  m  having  velocity  v  possesses  mv2/2  units 
of  kinetic  energy.  If  v  is  in  feet  per  second  and  m  in  engineers' 
kinetic  units  f  (the  unit  mass  being  equal  to  32.2  pounds-mass), 
mv2/2  is  in  foot-pounds. 

In  estimating  potential  energy  due  to  gravity  a  horizontal 
reference  plane  must  be  chosen.  If  a  particle  weighs  W  pounds 
and  is  z  feet  above  the  reference  plane,  it  possesses  Wz  foot- 
pounds of  potential  energy. J 

Thus,  the  potential  energy  of  any  particle  of  water  moving 
with  the  stream  increases  or  decreases  with  the  height  of  the 
particle  above  datum,  and  its  kinetic  energy  increases  or  de- 
creases with  its  velocity.  It  is  instructive  to  consider  some- 
what definitely  how  these  changes  occur  in  the  case  of  steady 
flow. 

(1)  Any  small  portion  of  the  fluid  is  continually  receiving 
energy  from  certain  adjacent  particles  and  giving  up  energy  to 

*  The  theory  of  steady  flow  of  gases  is  treated  in  Appendix  A. 

f  Theoretical  Mechanics,  Art.  218. 

J  Strictly  it  should  be  said  that  this  potential  energy  is  possessed  by 
the  system  consisting  of  the  body  and  the  earth.  See  Theoretical  Mechanics, 
Art,  362. 

43 


44        THEORY  OF  ENERGY  APPLIED  TO  STEADY  FLOW. 

others.    Thus,  consider  a  small  body  of  the  water  between  two 
transverse  planes  (as  X,  Fig.  25).     This  body  X  is  acted  upon 

by  pressures  exerted  by  the  adjacent 
B  bodies  of  water  P  and  Q.  Let  Ff  be 
the  area  of  the  cross-section  between 
P  and  X,  and  pf  the  normal  pressure 
per  unit  area  exerted  across  this  sec- 
tion; let  F"  be  the  area  of  the  cross- 
section  between  X  and  Q,  and  p"  the  intensity  of  pressure 
in  this  section;  and  let  v',  v"  be  the  velocities  in  the  cross- 
sections  F',  F"  respectively.*  In  a  short  interval  of  time 
Jt  the  body  X  receives  an  amount  of  energy  p'F'v'M  by 
reason  of  the  positive  work  done  upon  it  by  the  total  pres- 
sure Frp'\  the  body  P  loses  an  equal  quantity  of  energy 
because  an  equal  amount  of  negative  work  is  done  upon 
it.  During  the  same  time  the  body  X  loses  an  amount  of 
energy  p"F"i/'dt  because  of  the  negative  work  done  upon  it 
by  the  pressure  p"F"',  the  body  Q  gaining  an  equal  quantity 
of  energy.  In  other  words,  during  the  time  At  the  body  X 
receives  from  P  a  quantity  of  energy  p'F'tfM,  and  gives  up  to 
Q  a  quantity  p"F"i/'M.  Since  F'tf  =F"v"J  these  two  amounts 
of  energy  will  be  equal  if  pf  =p"',  hence  in  that  case  the  body 
X  neither  gains  nor  loses  energy  by  reason  of  the  pressures, 
but  acts  as  a  transmitter  of  energy  f  from  P  to  Q.  If,  how- 
ever, p'  and  p"  are  unequal,  the  body  X  receives  from  P  either 
more  or  less  energy  than  it  gives  to  Q.  But  in  any  case,  so 
far  as  the  process  here  considered  is  concerned,  one  portion  of 
water  gains  exactly  as  much  energy  as  other  parts  lose;  the 
energy  possessed  by  the  whole  stream  remains  constant,  while 
that  of  individual  particles  continually  varies. 

(2)  If  two  adjacent  portions  of  the  water  slide  over  one 
another,  the  tangential  forces  which  resist  such  sliding  result 
in  a  twofold  energy  changs.  So  far  as  the  two  portions  have 

*  The  velocity  is  assumed  uniform  throughout  each  cross-section.  If  this 
is  not  true,  the  reasoning  still  holds  for  an  elementary  portion  of  the  stream. 

f  This  is  analogous  to  the  transmission  of  energy  from  one  pulley  to  another 
through  a  belt. 


ENERGY   PASSING  ANY   GIVEN   SECTION.  45 

a  common  component  of  motion  parallel  to  the  sliding,  there 
is  a  transfer  of  energy  from  one  portion  to  the  other  exactly 
similar  to  that  just  explained  as  due  to  normal  pressure.  But 
.since  the  two  portions  move  unequally  in  the  stated  direction, 
the  energy  lost  by  one  is  not  equal  to  that  gained  by  the  other, 
The  negative  work  done  on  one  portion  is  always  greater  than 
the  positive  work  done  on  the  other,  so  that  the  net  result  is 
a  loss  of  energy.  The  mechanical  energy  thus  lost  is  trans- 
formed into  molecular  energy  or  heat.  This  transformation  of 
mechanical  energy  into  heat  is  called  dissipation  of  energy. 
Energy  thus  dissipated  is  practically  lost,  so  far  as  the  possi- 
bility of  its  utilization  is  concerned. 

(3)  Particles  adjacent  to  the  surface  of  the  pipe  (or  what- 
ever body  encloses  the  stream)  lose  energy  by  reason  of  the 
negative  work  done  upon  them  by  the  frictional  forces  exerted 
by  the  pipe  surface.  The  energy  thus  lost  is  dissipated  into  heat. 

It  is  thus  seen  that  in  any  steady  stream  there  is  in  general 
a  continual  transference  of  energy  in  the  direction  of  the  flow, 
accompanied  by  a  dissipation  of  mechanical  energy  into  heat. 
Because  of  this  dissipation,  the  energy  transferred  across  a  sec- 
tion A  (Fig.  25)  is  always  greater  than  that  transferred  across 
a  section  B  down-stream  from  A.  This  will  be  made  definite 
in  the  following  article. 

59.  Energy  Passing  Any  Given  Section.  —  A  useful  expression 
may  be  deduced  for  the  total  quantity  of  energy  passing  any 
cross-section  of  a  steady  stream  in  a  given  S 

time.  ^r^JZ    ["~f~^~-» 

Consider  any  section  of  the  stream,  as 


Let  p  =  intensity  of  pressure  at  every  point  in  the  section 

(Ibs.  per  sq.  ft.); 

J^=area  of  cross-section  (sq.  ft.); 

2=ordinate  of  centroid  of  section  (ft.)  above  a  horizon- 
tal datum  plane,  taken  as  reference  plane  in  esti- 
mating potential  energy; 
v  =  velocity  of  flow  across  the  section  (ft.  per  sec.); 


46        THEORY  OF  ENERGY  APPLIED  TO  STEADY  FLOW. 

W=  weight  of  water  passing  the  section  per  unit  time 

(Ibs.  per  sec.)  ; 

w  =  weight  of  unit  volume  of  water  (Ibs.  per  cu.  ft.) ; 
f)      W 
X     ~=  volume  of  water  passing  the  section  per  unit  time 

(cu.  ft.  per  sec.). 

Energy  passes  the  section  S  in  two  ways :  (a)  Each  particle 
of  water  passing  the  section  possesses  a  certain  quantity  of 
energy,  part  potential  and  part  kinetic;  (6)  the  particles  on 
one  side  of  the  section  are  at  every  instant  giving  up  energy  to 
the  particles  on  the  other  side  by  reason  of  the  pressure  and 
motion. 

(a)  During  a  short  interval  of  time  At,  WAt  pounds  of  water 
pass  the  section.  This  water  possesses  potential  energy  of 


amount  WAt-z,  and  kinetic  energy  of  amount  — ~ — >   i-e->  a 
total  quantity  of  energy 


(6)  Designating  by  X  and  Y  the  bodies  of  water  adjacent 
to  the  section  and  separated  by  it  (Fig.  26),  it  is  seen  that,  by 
reason  of  the  pressure  exerted  by  X  and  Y  upon  each  other, 
the  body  X  loses  and  the  body  Y  gains  a  quantity  of  energy 
which  may  be  computed  as  follows:  The  total  pressure  resist- 
ing the  motion  of  X  is  Fp'}  an  equal  and  opposite  force  acts 
upon  Y  in  the  direction  of  the  motion.  In  a  time  At,  the 
bodies  upon  which  these  forces  act  move  a  distance  vAt.  The 
force  Fp  acting  upon  X  does  an  amount  of  work  —  FpvAt,  and 
the  force  Fp  acting  upon  Y  does  an  amount  of  work  +  FpvAt. 
That  is,  X  loses  a  certain  amount  of  energy  and  Y  gains  an 
equal  amount;  or  there  is  a  transfer  from  X  to  Y  of  an  amount 
of  energy 

FpvAt. 

Since  W =  wFv,  the  quantity  of  energy  passing  the  section  in 
time  At  by  reason  of  the  transfer  of  energy  from  X  to  Y  may 


GENERAL  EQUATION  OF  ENERGY   IN   STEADY  FLOW.       47 
be  written  in  either  of  the  equivalent  forms 


w 


Combining  the  values  found  in  paragraphs  (a)  and  (6), 
there  results  for  the  total  energy  passing  the  section  S  in  the 
time  At  the  value 


w 


The  quantity  of  energy  passing  the  section  per  unit  time  is 
evidently 


w 


Stated  in  still  another  way,  it  is  seen  that  for  every  unit 
weight  oj  water  passing  any  section  a  quantity  of  energy 


y    w 

passes  the  same  section.  The  quantities  of  water  passing  dif- 
ferent sections  in  any  time  are  equal,  but  the  quantities  of 
energy  are  unequal,  because  p  and  v  in  general  change  from 
section  to  section. 

60.  General  Equation  of  Energy  in  Steady  Flow.  —  An  im- 

portant equation,  often  called  Bernoulli's  theorem,  is  obtained 
by  applying  the  theory  of  energy  to  a  portion  of  a  steadily 
flowing  stream. 

Referring  to  Fig.  25,  consider  the  volume  included  between 
two  fixed  cross-sections  A  and  B.  Let  pressure,  velocity,  area 
of  cross-section,  height  above  datum,  etc.,  be  represented  by 
the  same  symbols  as  above,  with  suffix  d)  for  the  up-stream 
section  A  and  (2)  for  the  down-stream  section  B. 

If  the  flow  is  steady,  it  is  evident  that  the  total  quantity 
of  mechanical  energy  contained  in  the  volume  AB  remains 


48        THEORY  OF  ENERGY  APPLIED  TO  STEADY  FLOW. 

constant.  For,  considering  any  elementary  volume  in  a  fixed 
position  in  the  stream,  the  water  occupying  it  at  any  instant 
has  the  same  mass  and  velocity  as  that  which  has  replaced  it 
at  any  succeeding  instant,  and  the  two  elements  therefore  pos- 
sess equal  quantities  of  energy,  both  kinetic  and  potential. 
The  total  mechanical  energy  gained  by  the  volume  AB  during 
any  time  must  therefore  equal  the  total  mechanical  energy 
lost  by  it  during  the  same  time. 

Consider  the  energy  gained  and  lost  by  the  volume  AB 
while  W  pounds  of  water  pass  any  section  of  the  stream. 

The  only  gain  of  mechanical  energy  is  that  passing  the  sec- 
tion A\  its  value  is 


The  mechanical  energy  lost  is  made  up  of  two  parts,  —  that 
passing  the  section  B}  and  that  dissipated  into  heat.  The  value 
of  the  former  part  is 


w 


The  value  of  the  energy  dissipated  will  be  represented  by  K. 

Equating  the  total  energy  gained  by  the  volume  AB  to  the 
total  energy  lost  by  it,  we  have 


-a-  .....   0) 


TT 

in  which  H7  =  ^. 

Equation  (1)  expresses  Bernoulli's  theorem,  or  the  general 
equation  of  energy  for  steady  flow. 

61.  Meaning  of  "  Head." — The  word  head  is  used  by  writers 
on  Hydraulics  in  a  somewhat  indefinite  way.    In  all  cases  it 


EFFECTIVE  HEAD.  49 

means  the  height  of  a  column  of  water,  either  actual  or  ideal. 
Thus,  "head  on  an  orifice/'  or  on  any  point  of  an  orifice,  has  in 
preceding  discussions  been  used  to  designate  the  vertical  height 
of  the  free  surface  of  water  above  the  point  under  consideration. 

Most  writers  at  the  present  time  use  the  word  head  in  the 
following  senses  : 

The  pressure  head  at  any  point  in  a  body  of  water  is  the 
height  of  a  column  of  water  which,  in  equilibrium,  would  by 
its  weight  produce  the  pressure  existing  at  the  point.  The 
pressure  head  corresponding  to  a  pressure  of  intensity  p  is 
therefore  equal  to  p/w. 

The  velocity  head  at  any  point  of  a  stream  is  the  height 
through  which  a  body  must  fall  from  rest  under  gravity  to 
acquire  a  velocity  equal  to  that  existing  at  the  given  point. 
The  velocity  head  corresponding  to  a  velocity  v  is  therefore 

v2/2g. 

The  sum  of  the  pressure  head  and  the  velocity  head  at  any 
point  of  a  stream  is  often  called  the  effective  head  at  that  point. 
It  seems  desirable,  however,  to  modify  the  meaning  of  this 
term  in  the  following  manner. 

62.  Effective  Head.—  It  has  been  shown  (Art.  59)  that  the 
total  energy  passing  a  given  section  of  a  steady  stream,  per  unit 
weight  of  water  discharged,  is  equal  to 


g    w 

Each  of  the  three  terms  in  this  expression  represents  a  linear 
magnitude,  and  each  may  be  called  a  "head."  The  second  and 
third  terms  may  be  called  velocity  head  and  pressure  head  re- 
spectively, in  accordance  with  usage  as  explained  above.  The 
term  z  has  been  called  elevation  head,  and  also  potential  head. 
The  former  term  will  be  here  adopted. 

Effective  head  will  here  be  defined  as  the  sum  of  the  pressure 
head,  velocity  head,  and  elevation  head. 

It  seems  desirable  to  include  the  term  z  in  the  definition  of 
effective  head,  for  the  reason  that  this  term  has  equal  signifi- 


50        THEORY  OF  ENERGY  APPLIED  TO  STEADY  FLOW. 

cance  with  the  other  two  in  estimating  the  total  energy  deliv- 
ered at  a  given  point  in  the  stream. 

The  symbol  H  will  hereafter  be  employed  to  designate  the 
value  of  the  effective  heacl  at  any  point  of  a  stream.    That  is, 


2g    w 

The  values  of  H  at  different  cross-sections,  like  those  of  z,  p, 
and  v,  will  be  distinguished  by  suffixes. 

63.  Lost  Head.  —  The  general  equation  of  energy  (Art.  60) 
shows  that  the  effective  head  decreases  along  the  stream  in  the 
direction  of  the  flow.  Thus,  with  the  above  notation,  the  equa- 
tion may  be  written 


The  quantity  H'  is  a  linear  magnitude,  and  expresses  the 
amount  by  which  the  effective  head  decreases  from  A  to  B 
(Fig.  25)  ;  H'  may  therefore  be  called  the  loss  of  head  between 
A  and  B.  That  is, 

The  loss  of  head  between  any  two  sections  of  a  stream  is  defined 
as  the  amount  by  which  the  effective  head  at  the  up-stream  section 
exceeds  that  at  the  down-stream  section. 

The  reasoning  by  which  the  general  equation  of  energy  was 
deduced  shows  that  H'  has  an  important  meaning  as  energ}^. 
Since  H'  =  K/W,  in  which  K  denotes  the  energy  lost  by  dissi- 
pation per  unit  time  between  A  and  B,  and  W  the  weight  of 
water  discharged  (across  any  section)  per  unit  time,  it  is  seen 
that 

The  loss  of  effective  head  between  any  two  sections  of  a  steady 
stream  is  equal  to  the  energy  lost  by  dissipation  between  those 
sections  per  unit  weight  of  water  discharged. 

In  practical  applications,  the  value  of  the  lost  head  between 
two  sections  is  sometimes  found  by  determining  HI  and  H2  and 
taking  their  difference;  and  sometimes  its  value  is  estimated 
from  a  consideration  of  its  energy  meaning  as  just  explained. 


LOST  HEAD.  51 

It  is  instructive  to  devote  some  time  to  the  application  of 
the  equation  of  energy  on  the  assumption  of  no  loss  of  head. 
In  some  cases  the  results  have  practical  value,  the  actual  losses 
being  small.  In  other  cases  such  a  treatment  is  useful  only  in 
illustrating  general  principles. 

EXAMPLES. 

1.  In  a  horizontal  pipe  6  inches  in  diameter  water  flows  with  a  velocity 
of  12  ft.  per  sec.     At  a  section  A  the  mean  pressure  is  50  Ibs.  per  sq.  in., 
and  at  a  section  B  it  is  40  Ibs.  per  sq  in.     Compute  the  quantity  of 
energy  passing  each  of  these  sections  in  one  second,  estimating  potential 
energy  with  reference  to  a  datum  plane  18  ft.  below  the  axis  of  the  pipe. 

Ans.  18,950  foot-pounds  and  16,540  foot-pounds. 

2.  In  Ex.  1  what  is  the  value  of  the  lost  head  between  A  and  J5? 
In  which  direction  is  the  flow? 

3.  The  diameter  of  a  pipe  is  12  inches  at  a  section  A,  16  ft.  above 
datum,  and  8  inches  at  a  section  B,  8  ft.  above  datum.     At  A  the  velocity 
is  12  ft.  per  sec.  and  the  pressure  head  12.6  ft.     Assuming  no  dissipation 
of  energy  between  A  and  B,  compute  the  velocity  and  the  pressure  head 
at  B.  Ans.  Velocity  =  27  ft.  per  sec. ;  pressure  head  =  10.9  ft. 


CHAPTER  V. 

APPLICATION  OF    GENERAL  EQUATION    OF  ENERGY, 
NEGLECTING  LOSSES  BY  DISSIPATION. 

64.  General  Method. — Assuming  no  loss  of  energy  by  dissi- 
pation in  any  part  of  the  stream,  the  general  equation  of  energy 
may  be  written 

v2     p 
2  +  77-  H —  =H  =  constant. 

2g    w 

The  general  method  of  applying  this  equation  is  as  follows  : 

(1)  Choose  a  datum  plane.    This  fixes  the  value  of  z  at 
every  point  of  the  stream. 

(2)  Notice  at  what  points  the  velocity  is  known.    If  at  any 
point  the  cross-section  is  very  great  in  comparison  with  its 
value  at  other  points,  the  velocity  will  be  so  small  that  the 
velocity  head  at  that  point  may  be  neglected. 

(3)  Notice  at  what  points  of  the  stream,  if  any,  the  pressure 
is  known. 

(4)  Notice  whether  there  is  any  point  of  the  stream  at 
which  the  three  parts  of  the  effective  head — elevation  head, 
pressure  head,  and  velocity  head — are  all  known.       If  there  is, 
the  value  of  H  becomes  known  at  that  point,  and  therefore  at 
all  sections  of  the  stream. 

(5)  If  there  is  no  point  at  which  elevation,  pressure,  and 
velocity  are  all  known,  note  the  points  at  which  the  value  of  H 
can  be  expressed  with  the  use  of  the  fewest  unknown  quantities. 
By  writing  expressions  for  H  for  two  or  more  sections,  it  may  be 
possible  to  solve  the  resulting  equations  and  determine  the 

unknown  quantities. 

52 


OF  iv* 

'F- 
FLOW  FROM  A  RESERVOIR  THROUGH  AN  ORIFICE.        53 

65.  Flow  From  a  Reservoir  Through  an  Orifice.  —  Consider  a 
small  orifice  in  the  side  of  a  vessel  (Fig.  27),  from  which  a  jet 
is  discharged  into  the  atmosphere.  Let  the 
center  of  the  smallest  cross-section  of  the  jet 
(S)  be  at  a  distance  h  below  the  level  of  the 
free  surface  of  the  water  in  the  vessel. 

The  particles  of  water  approaching  the 
orifice  doubtless  come  from  various  parts  of 
the   reservoir;    but  it  will   appear  presently 
that  the  same  solution  of  the  problem  will  result  whatever 
point  in  the  reservoir  be  taken  as  a  point  of  the  stream. 

(1)  Choose  as  datum  the  horizontal  plane  through  the  cen- 
ter of  the  stream  at  S.     (Any  other  horizontal  plane  might  be 
chosen.) 

(2)  The  velocities  of  particles  within  the  reservoir  are  very 
small,  except  in  the  neighborhood  of  the  orifice.     For  any  point 
at  some  distance  from  the  orifice  the  velocity  may  therefore  be 
taken  as  zero  without  sensible  error. 

(3)  The  jet  being  surrounded  by  the  atmosphere,  the  pres- 
sure within  the  stream  at  S  is  atmospheric.     Atmospheric  pres- 
sure also  exists  at  the  water  surface  A.    At  points  within  the 
vessel  (except  near  the  orifice  where  the  particles  have  sensible 
velocities)  the  pressure  follows  the  hydrostatic  law;   that  is,  at 
a  depth  y  below  the  free  surface  the  pressure  exceeds  atmos- 
pheric by  wy. 

(4)  At  any  point  within  the  reservoir  at  which  the  velocity 
is  sensibly  zero  the  value  of  the  effective  head  is  known  imme- 
diately.   Consider  a  point  B  (Fig.  27).    Whatever  the  value  of 
z  for  this  point,  we  have  for  the  value  of  the  pressure  head  (call- 
ing atmospheric  pressure  p0) 


w     w 
The  velocity  head  being  sensibly  zero, 


w  w  w 


54 


EQUATION   OF  ENERGY  WITHOUT  LOSSES. 


That  is, 


w 


so  that  the  value  of  the  effective  head  at  all  points  becomes 
known. 

To  complete  the  solution,  let  v  denote  the  velocity  of  the 
jet  at  S.    The  values  of  elevation  head,  pressure  head,  and 

velocity  head  for  this  section  are  0,  —  ,  and  ^-;  that  is, 


rr 

ti  = rrr— . 

w     2g 


Equating  the  values  of  H, 


V2 


It  is  thus  seen  that,  neglecting  frictional  losses  of  energy, 
Torricelli's  theorem  (Art.  37)  is  a  direct  consequence  of  the 
theory  of  energy. 

66.  Case  of  Unequal  Pressures.  —  If  the  pressures  at  the  sur- 
face of  the  reservoir  and  at  the  section  S  of 
the  jet  are  unequal,  let  their  values  (per 
unit  area)  be  p\  and  p2  respectively.  Then 
for  the  point  B  (Fig.  28)  we  have 


FIG.  28. 
and  for  the  point  S 


Equating  these  values, 


£_£>+(*-*);    H^ 

w     w  w 


rr          i, 

ti  = r;r— . 

w     2g 


w      w 


which  agrees  with  Art.  52. 


VELOCITY  OF  APPROACH.  55 

67.  Velocity  of  Approach. — If  the  cross-section  of  the  reser- 
voir is  not  so  great  that  the  downward  velocity  of  the  water  may 
be  neglected,  another  term  enters  the  formula. 

Let  vr  =  velocity  of  water  at  point  B  (Fig.  27); 

v  =  velocity  of  jet  at  section  S. 

Then,  assuming  atmospheric  pressure  at  A  and  at  S,  we  have 
for  the  point  B 

~w  2g' 

and  for  the  point  S 

w  •   2g" 

Equating  these  values  of  H, 

v'2     v2 


Let  F  =  cross-section  of  jet  at  8', 
Ff  =  cross-section  of  vessel  at  B. 

F'v'==Fv,     or     v'  =  (j^v. 

Substituting  this  value  of  t/  in  the  foregoing  equation  and  solv- 
ing for  v,  we  have 


If  the  pressures  at  A  and  S  are  unequal,  the  above  reasoning 
ieads  to  an  equation  similar  to  the  above  with  h4-p\/w—pz/w 
substituted  for  h. 

EXAMPLES. 

1.  If  the  diameter  of  the  jet  is  0.5"  and  that  of  the  vessel  2",  both 
being  circular  in  cross-section,  what  percentage  of  error  is  introduced 
by  neglecting  velocity  of  approach?  Ans.  About  0.2  per  cent. 

2.  For  what  value  of  the  ratio  F/F'  will  an  error  of  one  per  cent  be 
introduced  by  neglecting  velocity  of  approach? 


56 


EQUATION  OF  ENERGY  WITHOUT  LOSSES. 


68.  Effect  of  Velocity  of  Approach  in  Case  of  Rectangular 
Weir. — The  foregoing  discussion  has  referred  to  an  orifice  so 
small  that  no  important  error  results  from  regarding  the  head 
on  the  center  as  applying  to  all  parts  of  the  orifice.  The  effect 
of  velocity  of  approach  in  case  of  large  vertical  orifices  needs 
special  consideration.  It  will  be  sufficient  to  consider  a  rect- 
angular orifice,  since  it  is  this  form  that  possesses  most  prac- 
tical importance. 

Consider  a  stream  flowing  horizontally  in  an  open  channel 
(Fig.  29)  and  discharging  at  the  end  of  the  channel  through  a 


FIG.  29. 

rectangular  orifice  of  width  6,  the  upper  and  lower  edges  being 
at  depths  hi  and  h%  respectively  below  the  surface  of  the  water 
in  the  channel.  Consider  the  ideal  case  in  which  there  is  no 
contraction  of  the  stream  and  no  loss  of  energy  by  dissipation, 
and  assume  that  the  velocity  of  the  approaching  stream  has 
the  same  value  throughout  all  parts  of  any  given  cross-section. 
Let  ?/  be  the  value  of  this  velocity. 

At  the  cross-section  S  of  the  issuing  stream,  let  v=  velocity 
at  a  point  C  whose  depth  below  the  water  surface  in  the  chan- 
nel is  x.  The  value  of  v  is  found  by  applying  the  equation  of 
energy  to  the  point  C  and  a  point  B  of  the  approaching  stream. 

Let  z\  and  22  denote  the  ordinates  of  B  and  C  respectively 
above  any  assumed  datum;  then  the  depth  of  B  below  the 
water  surface  is  x+z^—zi,  and  the  pressure  head  at  B  is 

x  +22-2i+— , 
w' 

pQ  being  atmospheric  pressure.     Hence 


DISCHARGE   FROM  RESERVOIR  THROUGH  PIPE.  57 

For  the  point  C,  the  pressure  being  that  of  the  atmosphere, 
we  have 


w 
Equating  these  values  of  H, 


v2     v'2  , 


Taking  an  elementary  area  of  length  b  and  width  dx,  we 
have  for  the  discharge  per  unit  time  through  that  element 

dq=bvdx=b  (v'2  +  2gx)  *dx. 
Integrating, 


Inspection  of  this  result  shows  that  the  velocity  of  approach, 
so  far  as  its  effect  on  the  rate  of  discharge  is  concerned,  is 
equivalent  to  an  increase  of  v'2/2g  in  the  head  on  every  part 
of  the  orifice. 

In  case  of  a  weir,  the  value  of  hi  is  zero.  If  H  denotes  the 
"head  on  the  crest,"  i.e.,  the  depth  of  the  bottom  edge  (or 
crest)  of  the  orifice  below  the  surface  of  the  approaching  stream, 
and  hf  is  written  for  v'2/2g  (the  head  "  equivalent  to"  the  veloc- 
ity of  approach  vf),  the  formula  becomes 

q  =  %V2g-b[(H+h'f-h'*]. 

69.  Discharge  from  Reservoir  Through  Tube  or  Pipe.  —  If  a 

pipe  or  tube,  or  any  system  of  pipes,  leads  from  a  reservoir  and 
discharges  a  stream  at  any  point,  it  is  evident  that  the  velocity 
of  discharge  (on  the  assumption  that  no  energy  is  dissipated) 
is  given  by  the  same  formula  as  that  applying  to  discharge  from 
an  orifice. 


B 


58  EQUATION  OF  ENERGY  WITHOUT  LOSSES. 

Thus,  let  any  system  of  pipes  lead  from  a  large  reservoir 

(Fig.  30)  and  discharge  into  the 
atmosphere  at  a  point  whose 
depth  below  the  water  surface 
is  h.  The  reasoning  employed 
in  the  discussion  of  discharge 
through  an  orifice  applies  un- 

r  IG.  30.  1111  • 

changed,    and     the    velocity    of 
the  stream  at  S  is  given  by  the  formula 

v=V2gh. 

If  the  cross-section  of  the  issuing  stream  is  known,  the  rate 
of  discharge  is  known  from  the  relation 

q=Fv. 

70.  Pressure  at  Any  Section. — After  the  value  of  q  has  been 
determined  the  velocity  and  pressure  at  any  point  in  the  stream 
may  be  computed,  if  the  cross-section  at  that  point  is  known. 

Thus,  in  Fig.  30,  let  h  =40  ft.,  and  let  the  diameter  of  the 
jet  at  S  be  1  inch.  Let  it  be  required  to  determine  the  velocity 
and  pressure  at  a  point  C,  10  ft.  higher  than  S,  the  pipe  at  C 
being  1.25  inches  in  diameter. 

Let  v2=  velocity  at  S;  Vi  =  velocity  at  C;  pi  =  pressure  at 
C.  As  above,  we  have 

!-*-«. 

Taking  datum  plane  through  the  center  of  the  jet  at  S,  we  have 

w     2g     w 
For  the  point  C, 


w     2g 
Equating  values  of  H, 


w 


PRESSURE  CAN  NEVER  BE  NEGATIVE.  59 


Pi    Po  ,  qn    ^ 

or  —  =  --  r  oU  —  pr~. 

w      w  2g 

/5\  2  v-i2      /4\  4  v%2 

But          vi=v2,     or          =         '      =  16A'    hence 


That  is,  the  pressure  at  C  exceeds  that  of  the  atmosphere  by 
the  equivalent  of  13.6  ft.  head. 

It  will  be  noticed  that,  if  different  points  of  the  stream  at 
the  same  level  be  compared,  the  pressure  is  greatest  where  the 
velocity  is  least.  Since  a  contraction  of  the  stream  increases 
the  velocity,  it  causes  a  decrease  of  pressure. 

More  generally,  consider  what  is  implied  by  the  equation 

J)        V^ 

z  +  —  -\-JT-  =H  =  constant. 
w    2j 

(1)  For  a  stream  of  uniform  cross-section  and  consequently 
having  equal  velocities  at  all  sections,  the  pressure  head  in- 
creases as  the  elevation  decreases,  and  vice  versa;  the  variation 
of  pressure  follows  the  hydrostatic  law. 

(2)  For  a  stream  all  parts  of  which  are  at  the  same  level 
the  pressure  head  increases  as  the  velocity  head  decreases,  and 
vice  versa. 

When  losses  of  energy  by  dissipation  are  considered,  these 
principles  are  greatly  modified.  But  it  is  instructive  to  con- 
sider carefully  the  conditions  in  the  ideal  case  of  no  loss  of 
energy. 

71.  Pressure  Can  Never  be  Negative.  —  In  such  a  case  as  that 
shown  in  Fig.  30,  the  velocity  of  the  jet  at  £  is  independent 
of  the  dimension's  or  elevation  of  any  part  of  the  pipe  leading 
from  the  reservoir  to  the  point  of  discharge.  The  only  condi- 
tions affecting  the  velocity  of  outflow  are  the  pressure  and  ele- 
vation at  A  and  S,  and  the  relation  between  the  cross-sectional 
areas  at  these  points.  Between  these  sections  the  pipe  may 


60  EQUATION  OF  ENERGY  WITHOUT  LOSSES. 

vary  in  diameter  in  any  way,  and  its  elevation  may  vary  in 
any  way  without  affecting  the  velocity  of  discharge.     There  is, 
however,  an  important  limitation  on  the  application  of  this 
theory,  even  without  the  consideration  of  losses  of  energy. 
From  the  equation 

n     v2 

z+ — \-JT  =H=  constant 
w     2g 

it  is  evident  that,  by  increasing  v  or  2,  the  pressure  p  may  be 
decreased  to  any  assigned  value.  The  velocity  of  outflow,  and 
therefore  the  rate  of  discharge,  being  fixed,  if  at  any  point  the 
cross-section  is  made  very  small  the  velocity  becomes  very 
great.  Since  v=q/F,  there  is  no  limit  to  the  value  which  may 
be  given  to  v  by  decreasing  F.  For  any  value  of  2,  therefore, 
p/w  may  be  made  zero,  or  even  negative,  by  decreasing  the 
cross-section  of  the  pipe. 

A  negative  pressure  would  mean  a  tension,  and  the  nature 
of  a  liquid  is  such  that  a  tendency  to  tension  causes  separation 
of  the  particles.*  Therefore  if  the  pressure  at  any  point  of 
the  stream,  as  computed  by  the  foregoing  theory,  is  found  to 
be  negative,  the  conclusion  to  be  drawn  is  that  the  stream  will 
break  at  that  point,  and  the  solution  must  be  modified. 

If,  at  the  point  where  the  pressure  as  computed  from  the 
formula  has  the  least  (algebraic)  value,  this  value  is  negative, 
the  actual  pressure  may  be  put  equal  to  zero;  and  the  velocity 
computed  on  this  assumption  may  be  regarded  as  the  greatest 
possible  value  of  the  velocity  at  that  point  on  the  supposition 
of  no  dissipation  of  energy. 

EXAMPLES. 

1,  In  Fig.  31,  let  the  diameter  of  the  pipe  have  the  following  values 
at  different  points:   at  B,  3";  at  C,  2.5";  at  D,  2.5";   at  E,  2".     Let 
the  diameter  of  the  jet  at  F  be  1.75".     Determine  the  velocity  and 
pressure  at  each  of  the  four  points  B,  C,  D,  E. 

2.  Keeping  the  size  of  the  pipe  unchanged,  how  high  can  the  pipe 
be  carried  at  D  without  reducing  the  pressure  to  absolute  zero? 

*  Strictly,  it  should  be  said  that  under  certain  conditions  a  slight  tension 
can  exist  in  liquids;  but  the  statement  above  made  is  practically  true. 


EXAMPLES. 


61 


3.  With  the  pipe  D  at  the  elevation  shown  in  Fig.  31,  how  small  may 
the  diameter  be  without  reducing  the  pressure  to  absolute  zero?  What 
will  happen  if  the  pipe  is  made  smaller  than  this  limiting  size? 

Ans.  Diameter  at  D  =  1.80". 


FIG.  31. 

4o  What  will  be  the  velocity  of  discharge  from  the  siphon  shown  in 
Fig.  32?  Ans.  11.3  ft.  per  sec. 

5.  Discuss  the  pressure  throughout  the  tube  in  Fig.  32,  assuming 
the  diameter  to  be  uniform.  How  small  may  the  pipe  be  at  A  (as  com- 
pared with  its  size  at  B)  without  making  the  pressure  at  A  absolutely 
zero?  Diameter  at  A 

Ans-  Diameter  at  B  =  0-51  (llmltmS  value)' 


FIG.  32. 


_l l_ 


FIG.  33. 


6.  In  Fig.  33,  let  the  diameter  of  the  pipe  be  1"  and  that  of  the  jet 
at  B  1.75".    Compute  the  pressure  at  the  point  A.     At  what  points  in 
the  pipe  has  the  pressure  the  least  and  greatest  values? 

Ans.  At  A,  p/w  =  p0/w-4.3. 

7.  In  Fig.  34,  let  the  diameter  of  the  diverging  tube  be  1"  at  B  and 
1.5"  at  Cj  the  jet  at  S  having  the  same  size  as  the  tube  at  C.    Compute 
velocity  and  pressure  at  B  and  at  C. 

Ans.  At  B,  v=  51.2;  p/w=p0/w-32.5. 

8.  In  Fig.  34,  if  the  tube  were  cut  off  at  B,  how  would  the  discharge 
be  affected? 


62 


EQUATION   OF  ENERGY  WITHOUT  LOSSES. 


9.  In  Fig.  34,  what  is  the  effect  of  increasing  the  diameter  of  the 
tube  at  C,  that  at  B  remaining  unchanged?     What  is  the  greatest  pos- 
sible value  of  the  velocity  at  B?  Ans.  51.9  ft.  per  sec. 

10.  In  Fig.  35,  let  the  diameter  of  the  pipe  be  2"  at  A  and  1"  at  B, 
its  axis  being  horizontal.     The  pressures  at  the  two  sections  are  meas- 
ured by  the  heights  of  water  standing  in  the  vertical  tubes  A  A'  and  BB'. 
If  the  water  surface  is  2'  higher  at  Af  than  at  B',  what  is  the  rate  of 
discharge?  Ans.  9  =  .064  cu.  ft.  per  sec. 

11.  Fig.  36  represents  a  pipe  in  which  water  flows  in  the  direction 
AB.     In  order  to  measure  the  difference  between  the  pressures  at  the 


FIG.  35. 

sections  A  and  B,  tubes  A  A'  and  BB'  are  inserted  at  the  two  sections 
and  connected  with  the  two  branches  of  a  U  tube  M ,  containing  mercury. 
If  mercury  fills  the  portion  of  the  tube  A'B'  and  water  the  portions  AA' 
and  BB',  the  difference  between  the  heights  of  A'  and  B'  indicates  the 
difference  between  the  pressures  in  the  sections  A  and  B.  The  specific 
gravity  of  mercury  may  be  taken  as  13.6. 

(a)  If  the  vertical  height  of  B'  above  A'  is  h,  what  is  the  difference 
"between  the  pressures  in  the  tube  at  A'  and  B'}  expressed  in  "head' 
or  height  of  water  column?  Ans.  13. 6A. 

(6)  What  is  the  difference  between  the  pressure  at  B'  and  that  at  A"} 
on  a  level  with  B"!  Ans.  12 M. 

(c)  What  is  the  difference  between  the  values  of  the  pressure  head 
at  C  and  D,  these  points  being  at  the  same  level*? 

12.  Wit^h  the  arrangement  shown  in 
Fig.  36,  let  the   diameter   at  A  be  3" 
and  that  at  B  2".    If  h  =  6",  what  is  the 
velocity  e>f  flow  at  5? 

Ans.  22.5  ft.  per  sec. 

13.  In  Fig.  36,  let  the  cross-sectional 
a'reasfat  A   and   B  be   Ft  and  F2,  and 
the.  corresponding  velocities  Vi  and  v0, 

and  let  h  be  the  difference  of  level  of  the  mercury  -surfaces  at  A' 
and  B'.  Deduce  the  following  formulas  (s  denotes  specific  gravity  of 
mercury) : 


A" 

-r— 

B' 

< 

__j_ 

A' 

M 

C 

- 

A 

\ 

R 

->- 

FIG.  36. 


EXAMPLES.  63 

\2(s-l)gh 


14.  In  Fig.  35,  let  the  diameters  at  A  and  B  be  4"  and  3"  respectively 
and  suppose  pistons  to  be  fitted  into  the  pipe  at  the  two  sections,  the 
intervening  space  being  filled  with  water.  If  the  piston  at  A  advances 
at  the  rate  of  6  ft.  per  sec.,  and  the  height  of  A'  above  the  center  of  the 
pipe  is  7',  compute  the  difference  between  the  total  pressures  upon  the 
two  pistons.  Neglect  friction. 

Ans.  75  Ibs.,  neglecting  atmospheric  pressure. 


CHAPTER  VI. 

APPLICATION  OF  GENERAL  EQUATION  OF  ENERGY, 
TAKING  ACCOUNT  OF  LOSSES. 

72.  Methods   of  Estimating   Loss   of  Head. — When     losses 
of  energy  by  dissipation  are  taken  into  account,  it  is  necessary 
to  estimate  the  value  of  H'  in  the  general  equation  of  energy, 

H  i  — HZ =H  . 

As  explained  in  Art.  63,  this  term  is  called  the  loss  of  head 
between  the  two  sections  at  which  the  effective  head  has  the 
values  Hi,  H2}  respectively,  the  former  referring  to  the  up- 
stream section.  The  energy  meaning  of  H',  as  stated  in  Art. 
63,  should  be  kept  clearly  in  mind.  It  is  the  energy  lost  by 
dissipation  between  the  two  sections  considered,  per  pound  of 
water  discharged. 

The  value  of  H'  in  particular  cases  may  be  estimated  either 
(a)  theoretically  or  (b)  experimentally. 

(a)  If  the  loss  of  energy  occurring  between  the  two  sections 
can  be  estimated  theoretically,  the  value  of  H'  can  be  com- 
puted from  its  energy  meaning  as  just  stated. 

(b)  If  the  values  of  HI  and  H2  can  be  determined  by  actual 
measurement,  the  value  of  Hf  becomes  known  from  the  equa- 
tion H'  =  Hl-H2. 

In  most  cases  in  which  the  first  method  is  employed  the 
theoretical  value  of  H'  involves  coefficients  whose  values  can 
be  known  only  by  experiment. 

73.  Experimental  Determination  of  Lost  Head. — The  deter- 
mination of  the    effective  head  at  any  section  requires   the 

64 


WATER   PIEZOMETER. 


65 


measurement  of  z,  p,  and  v.  The  elevation  above  datum  may 
be  found  by  direct  measurement  or  leveling.  It  remains  to 
consider  how  pressure  and  velocity  may  be  determined. 

The  mean  velocity  in  any  section  is  known  from  the  relation 
v  =  q/F,  as  soon  as  the  rate  of  discharge  is  known.  Methods 
of  measuring  q  will  be  discussed  in  Chapter  XIII.  If  the  tvvo 
cross-sections  to  be  compared  are  equal,  the  values  of  v  are 
equal,  and  need  not  be  known  in  order  to  determine  the  loss 
of  head  between  the  sections. 

The  pressure  at  any  section  of  a  pipe  may  be  determined 
by  tapping  the  pipe  and  attaching  a  pressure-gauge.  Several 
forms  of  pressure-gauges  are  used. 

74.  Water  Piezometer.  —  The  simplest  form  of  pressure- 
gauge  is  a  tube  inserted  in  the  pipe  at  right  angles  to  its  axis 
and  carried  up  to  such  height  as  may  be  necessary  to  prevent 
overflow,  the  top  being  open.  Thus,  in  Fig.  37,  the  pressure 
at  any  point  A  in  the  cross-section  S  is  equal  to  the  pressure 
due  to  the  height  of  the  column  of  water  AA'  plus  atmospheric 
pressure;  that  is,  the  pressure  at  any  depth  h  below  the  top  of 
the  column  is  po  +  wh,  this  law  holding  from  the  top  of  the 
column  to  the  bottom  of  the  cross-section  S.  That  it  holds 
throughout  the  cross-section  follows  from  the  fact  that  the 
direction  of  motion  of  the  water  is  at  right  angles  to  the  plane 
of  the  cross-section.  The  reasoning  of  Art.  11  applies  if  the 


S 
FIG.  37. 


points  A  and  B  (Fig.  3)  are  in  the  plane  of  the  cross-section 
since  the  elementary  prism  AB  has  no  acceleration  in  the  direc- 
tion of  its  length.* 

*  This  reasoning  is  rigorously  true  on  the  assumption  that  all  particles 


66 


EQUATION  OF  ENERGY  WITH  LOSSES. 


If  the  pipe  is  not  horizontal  (Fig.  38),  the  law  p  = 
still  holds  throughout  the  tube  and  the  cross-section  S,  h  being 
measured  vertically. 

It  obviously  makes  no  difference  whether  the  tube  com- 
municates with  the  cross-section  S  at  the  highest  point  or  at 
some  other  part  of  the  perimeter.  Thus,  if  Fig.  39  is  a  cross- 
sectional  view,  and  if  the  tubes  A  A'  and  BE'  are  inserted  at 
different  points,  the  tops  of  the  columns  (Af  and  Bf)  will  be  at 
the  same  level. 

In  order  that  the  piezometer  may  give  a  reliable  indication 
of  the  pressure  in  the  pipe,  the  axis  of  the  tube  at  the  point 


A  .. 


B 


C' 


FIG. 


of  attachment  must  be  at  right  angles  to  the  axis  of  the  pipe. 
Thus,  if  three  tubes  are  inserted,  as  at  A,  B,  and  C  (Fig.  40), 
the  friction  of  the  water  passing  the  orifice  will  increase  the 
pressure  in  the  tube  A  and  decrease  it  in  the  tube  C,  while 
the  column  BBf  will  indicate  the  pressure  correctly.  The  tops 
of  the  columns  will  stand  at  different  levels,  as  shown. 

The  foregoing  discussion  applies  only  to  the  case  in  which 
the  pressure  throughout  the  cross-section  of  the  pipe  is  greater 
than  that  of  the  atmosphere. 

If,  at  the  point  A  (Fig.  39),  the  pressure  within  the  pipe 
is  less  than  that  of  the  atmosphere,  no  column  of  water  will 

of  water  move  in  straight  lines  parallel  to  the  axis  of  the  pipe,  and  in  straight 
pipes  the  conclusion  undoubtedly  holds  within  the  limits  of  accuracy  ordinarily 
attainable  in  hydraulic  measurements.  For  the  most  accurate  work  it  is 
best  to  connect  the  piezometer  tube  with  a  chamber  communicating  with  the 
cross-section  of  the  pipe  by  small  orifices  at  several  points  of  the  circum- 
ference. 


MERCURY  GAUGE. 


67 


FIG.  41. 


be  sustained  in  the  tube,  but  the  external  pressure  will  con- 
tinually force  air  into  the  pipe  through  the  tube.  The  pressure 
may  in  such  case  be  measured  by  what  is  called  a  "vacuum 
piezometer  "  (Fig.  41).  The  tube  inserted 
at  A  is  carried  upward,  then  bent  down, 
the  end  dipping  into  an  open  vessel  of 
water  at  C.  The  pressure  within  the 
bent  tube,  after  a  condition  of  equilibrium 
has  been  established,  will  be  less  than 
atmospheric,  and  water  will  rise  from  the 
vessel  to  some  height  CC".  In  the  othei  — 
branch  of  the  tube,  water  will  stand  at 
some  height  A  A',  which  may  or  may  not  be  zero,  depending 
upon  the  initial  conditions. 

Neglecting  the  weight  of  the  air  in  the  tube,  the  pressure 
within  the  air-space  in  the  tube  is  uniform.  Therefore,  at  a 
vertical  distance  below  A'  equal  to  CC'  (in  the  column  AAf 
or  the  section  S),  the  pressure  is  atmospheric.  The  pressure 
at  any  point  in  the  section  S  can  therefore  be  determined  if 
the  tops  of  the  two  columns  A  A!  and  CC'  can  be  observed. 

75.  Mercury  Gauge. — For  high  values  of  the  pressure  the 
piezometer  tube  would  need  to  be  so  long  as  to  render  its  use 
impracticable.  In  such  cases  it  may  be  possible  *to  use  a 
mercury  gauge,  such  as  is  represented  in 
Fig.  42. 

The  tube  P  communicates  at  A  with 
the  cross-section  of  the  pipe  at  which 
it  is  desired  to  measure  the  pressure, 
and  also  communicates  with  the  reser- 
voir R  at  the  top.  This  reservoir  is 
partly  filled  with  mercury,  the  tube  P 
and  the  space  in  the  reservoir  above  the 
mercury  being  filled  with  water.  The 
vertical  tube  T,  open  at  both  ends,  is  fitted  tightly  into  the 
reservoir  at  the  top,  and  projects  far  enough  down  so  that  the 
lower  end  is  always  immersed  in  mercury. 


FIG.  42. 


68  EQUATION  OF  ENERGY  WITH  LOSSES. 

Throughout  the  space  above  the  mercury  in  the  reservoir, 
the  tube  P,  and  the  cross-section  S  of  the  pipe,  the  pressure 
varies  according  to  the  hydrostatic  law;  the  pressure  at  the 
mercury  surface  B  being  equal  to  that  at  points  at  the  same 
level  in  the  cross-section  S  or  in  the  tube.  If  the  pressure  at 
B  is  greater  than  atmospheric,  mercury  rises  in  the  tube  T 
until  the  pressure  at  B  due  to  the  mercury  column  BC,  plus 
atmospheric  pressure,  is  equal  to  the  pressure  at  the  same 
point  communicated  through  the  tube  P.  If,  by  means  of  a 
fixed  vertical  scale,  the  height  of  the  column  BC  can  be  meas- 
ured, the  pressure  at  B,  and  therefore  at  any  point  in  the  sec- 
tion S,  can  be  determined. 

Thus,  let  h  =  height  of  column  BC]  s  =  specific  gravity  of 
mercury;  p0  =  atmospheric  pressure;  p'  =  pressure  at  surface 
of  mercury  in  reservoir.  Then 


w     w 

If  p  denotes  the  intensity  of  pressure  at  a  point  in  the  cross- 
section  S  whose  vertical  distance  below  the  horizontal  plane 
through  B  is  x, 


WWW 

The  tube  P  must  be  kept  free  from  air,  for  which  purpose 
an  air-cock  should  be  provided  at  its  highest  point. 

76.  Bourdon  Gauge.  —  Where  precision  and  accuracy  are 
not  required  the  form  of  gauge  ordinarily  used  with  steam 
boilers  is  often  employed  for  the  measurement  of  water  pressure. 

EXAMPLES. 

1.  In  order  to  determine  the  loss  of  head  between  two  sections  A 
and  B  of  a  straight  pipe  of  uniform  cross-section,  piezometers  were 
attached  at  the  two  sections.  The  top  of  the  column  at  A  stood  150' 
above  a  horizontal  datum  plane,  and  that  of  the  column  at  B  140'  above 
the  same  plane.  The  center  of  the  pipe  at  A  was  80',  and  that  at  B  60', 
above  datum. 


LOSS  OF  HEAD  IN  STANDARD   ORIFICE.  69 

(a)  What  was  the  total  loss  of  head  between  A  and  B? 

(6)  What  was  the  pressure  at  the  center  of  each  of  the  cross-sections 
A  and  £? 

(c)  In  which  direction  was  the  flow?  (How  can  the  direction  of 
flow  always  be  determined  from  an  experiment  of  this  kind?) 

2.  With  the  pipe  as  in  Ex.  1,  suppose  the  piezometers  to  be  replaced 
by  mercury  gauges.     Let  the  surfaces  of  mercury  in  the  reservoirs  of 
the  two  gauges  be  82.63'  and  59.80'  respectively  above  datum,  and  let 
the  vertical  heights  of  the  mercury  columns  be  12.86'  and  13.46'  respect- 
ively. 

(a)  Compute  the  loss  of  head  between  A  and  B. 
(6)  Compute  the  pressure  at  the  center  of  each  of  the  two  sections. 

Ans.  (a)  14.67ft. 

3.  Take  data  as  in  Ex.  2,  except  that  the  size  of  the  pipe  is  not  uniform, 
(a)  What  additional  data  must  be  given  in  order  that  the  loss  of 

head  may  be  computed? 

(6)  If  the  diameter  is  12"  at  A  and  10"  at  B,  and  the  rate  of  dis- 
charge is  6  cu.  ft.  per  sec.,  what  is  the  loss  of  head  between  A  and  B 
when  the  gauges  read  as  stated  in  Ex.  2?  Ans.  (6)  13.70  ft. 

77.  Loss  of  Head  in  Standard  Orifice. — The  velocity  of 
efflux  from  a  standard  sharp-edged  orifice  (Art.  45)  is 

v  =  cfV2gh} 

in  which  the  coefficient  of  velocity  cf  has  a  value  fairly  well 
established  by  experiment. 

Comparing  the  values  of  the  effective 
head  at  two  points,  one  of  which  is  within 
the  reservoir  where  the  velocity  is  practically 
zero,  and  the  other  in  the  contracted  section 
of  the  iet  (as  the  points  B  and  S,  Fig.  43),  we 
have  FIG-  43' 


-     -- 


"T 


p0  being  atmospheric  pressure  and  v  the  velocity  of  flow  at 
section  S,  and  the  datum  plane  being  taken  at  the  level  of  the 


70  EQUATION  OF  ENERGY  WITH   LOSSES, 

center  of  the  orifice.     Hence 

,,2  i_c/2  V2 


The  value  of  Hf  is  thus  expressed  in  terms  either  of  the  head 
on  the  orifice  or  of  the  velocity  of  the  jet. 
If  c'  =  0.98  (Art.  45),  the  values  become 


, 


H'  =  0.0407t  =  0.041 


-V 


78.  Loss  of  Head  in  Short  Tube. — Consider  next  the  loss  of 
head  in  case  of  discharge  from  a  short  cylindrical  tube.  The 
length  is  assumed  to  be  just  sufficient  so  that  the  stream,  after 
converging  as  it  enters  the  tube,  expands  and  fills  the  tube  at 
the  outer  end  (Art.  38).  The  inner  edge  of  the  tube  is  sup- 
posed to  be  square  (Fig.  44). 


*— 


FIG.  44. 


FIG.  45. 


The  expression  for  Hr  in  terms  of  the  coefficient  of  velocity 
is  the  same  as  in  the  case  of  the  orifice;  but  the  value  of  this 
coefficient  is  different.  As  determined  experimentally  it  is 


which  gives 


c'  =  0.82  (nearly), 


'  =  0.33/1  =  0.49. 


If  the  tube  projects  within  the  vessel,  as  in  Fig.  45,  the 
velocity  of  the  jet  is  less  than  in  the  preceding  case.  Experi- 
ment gives 

^  =  0.72  (about) 


LOSS  OF  HEAD  IN  UNIFORM  STRAIGHT  PIPE.  71 

and 

i  v2 

#'=0.48/i  =  0.932-. 

79.  Loss  of  Head  in  Straight  Pipe  of  Uniform  Cross-section 

— The  loss  of  head  in  a  given  length  of  a  uniform  straight  pipe 
depends  upon  the  nature  of  the  inner  surface,  the  diameter, 
and  the  velocity  of  flow.*  Experiment  indicates  that  it  is 
independent  of  the  pressure  existing  in  the  pipe.  Since  loss 
of  head  is  proportional  to  energy  dissipated  (Art.  63),  it  obvi- 
ously depends  upon  the  friction  between  the  water  and  the  pipe, 
as  well  as  upon  the  friction  and  impact  of  the  particles  of  water 
among  themselves. 

No  theoretical  formula  has  been  proposed  which  can  be 
depended  upon  to  give  more  than  a  rough  approximation  to 
the  actual  losses  of  head  in  pipes.  The  following  formula, 
applicable  to  a  uniform  pipe  of  any  form  of  cross-section,  is 
often  employed,  and  is  the  basis  of  most  of  the  formulas  that 
will  be  considered  in  the  following  pages : 
Let  F  =  cross-sectional  area; 

C  =  length  of  perimeter  of  cross-section; 
F 

r=C> 
#'=loss  of  head  in  length  I  of  pipe; 

i>=mean  velocity  of  flow. 

Then  H'=f7Yg- 

In  this  formula  f  is  a  coefficient  whose  value  depends  upon 
the  roughness  of  the  surface  of  the  pipe,  and  also  (although 
to  a  less  degree)  upon  the  size  of  the  pipe  and  the  velocity  of 
flow.  The  quantity  r  is  a  length,  and  is  called  the  hydraulic 
ridius  of  the  cross-section;  its  value  depending  only  upon  the 
form  and  size  of  the  section. 

*  Doubtless,  also,  upon  the  temperature  of  the  water;  but  regarding  this 
little  experimental  knowledge  is  available. 


72  EQUATION  OF  ENERGY    WITH  LOSSES. 

The  theoretical  basis  of  this  formula  will  be  considered  in 
Chapter  IX. 

80.  Cylindrical  Pipe.  —  If  the  cross-section  of  the  pipe  is  a 
circle  of  diameter  d,  we  have 

nd2         F    d 


Hence  (writing  /  for  4/')  the  formula  becomes 


Experiment  indicates  that  the  coefficient  or  "friction  factor  " 
/  in  this  formula  is  not  constant  for  a  given  kind  of  pipe  but 
varies  with  both  d  and  v,  decreasing  as  each  of  these  quantities 
increases.  The  variation  with  v  appears,  however,  to  be  less 
important  than  that  with  d.  For  new  or  clean  cast-iron  pipe, 
and  for  velocities  within  the  range  ordinarily  occurring  in 
practice,  Darcy  recommended  a  formula  equivalent  to  the  fol- 
lowing (d  being  in  feet)  : 


For  old  pipe  these  values  should  be  doubled.* 

This  formula  may  be  used  in  solving  the  examples  which 
follow.  The  values  of  /  for  pipes  of  different  kinds  will  be 
considered  further  in  Chapter  IX. 

81.  Loss  of  Head  at  Entrance  to  Pipe.  —  If  water  is  con- 
ducted from  a  reservoir  by  a  straight  pipe,  the  conditions  of 
flow  are  less  uniform  near  the  entrance  to  the  pipe  than  they 
become  a  little  farther  on.  The  contraction  and  expansion  of 
the  enterirg  stream  result  in  a  loss  of  head  near  the  entrance 

*  Recherches  expe>imentales  relatives  au  mouvement  de  1'eau  dans  bs 
tuyeaux.  Chapter  IV,  also  p.  228. 


LOSS  DUE  TO  SUDDEN  ENLARGEMENT  OF  PIPE.          73 

that  is  much  greater  than  occurs  in  an  equal  length  where  the 
flow  has  become  uniform.  The  entrance  loss  is,  however, 
small  in  comparison  with  the  total  loss  unless  the  pipe  is  short, 
and  for  very  long  pipes  it  may  be  neglected. 

For  those  cases  in  wilich  entrance  loss  is  important  enough 
to  be  considered,  a  formula  for  it  is  obtained  by  assuming 
that  the  conditions  in  the  entrance  portion  of  the  pipe  are 
similar  to  those  existing  in  a  short  tube  discharging  into  the 
atmosphere.  It  is  assumed  that,  for  any  given  pipe,  the  loss 
of  head  at  entrance  varies  mainly  with  the  velocity  of  flow, 
and  that  it  has  the  same  value  as  for  a  short  tube  of  the  same 
diameter,  provided  the  conditions  are  such  that  the  velocities 
are  equal  in  the  tube  and  the  pipe.  The  entrance  loss  may 
therefore  be  expressed  in  terms  of  the  velocity  by  the  formulas 
already  given  (Art.  78)  for  the  case  of  a  short  tube.  Practi- 
cally, the  following  may  be  taken  as  sufficiently  correct : 

v2 
H'  =  .5  -y-  for  pipe  not  projecting  into  reservoir; 

v2  * 
H  =2~  for  projecting  pipe. 

82.  Loss  of  Head  Due  to  Sudden  Enlargement  of  Pipe. — 
If  the  cross-section  of  a  pipe  enlarges  suddenly  from  an  area  FI 
to  an  'area  F2  (Fig.  46),  the  irreg- 
ular motions  of  the  particles  of 
water  due  to  the  sudden  expansion 
of  the  stream  cause  a  dissipation  of 
energy  and.  consequent  loss  of  head. 
If  vi  and  v2  denote  the  values  of  the  FlG  46 

velocity    at    the    two    sections,    the 

loss  of  head  is  found  by  experiment  to  be  given  approximately 
by  the  formula 

H,_(vi-v2)2 
Since  FIV\  =F2v2,  this  formula  may  be  written 


74 


EQUATION  OF  ENERGY  WITH  LOSSES. 


2g- 

Special  Case. — If  the  ratio  F\/Fz  is  very  small,  the  above 
formula  reduces  to 

77'  -V]2 

~  9 ' 

This  applies  to  the  case  in  which  a  pipe  discharges  into  a  reser- 
voir at  a  point  below  the  water  surface. 

83.  Loss  of  Head  Due  to  Sudden  Contraction  of  Stream.— 
If  the  cross-section  of  the  pipe  decreases  suddenly  in  the  direc- 
tion of  the  flow  (Fig.  47),  there  is 
a  loss  of  head,  but  less  than  in  the 
case  of  sudden  enlargement.  The 
value  of  the  loss  depends  upon  the 
ratio  of  the  two  cross-sections  in  a 
way  that  can  be  known  only  by 
experiment.  Let  F\  and  F2  denote 

the  larger  and  smaller  cross-sections  respectively,  Vi  and  v2  being 
the  corresponding  velocities;  then  the  loss  of  head  may  be 
expressed  by  the  formula 


FIG.  47. 


_ 

k  being  an  experimental   coefficient  depending  upon  F2/Fi. 
The  following  values  of  k  are  based  upon  data  given  by  Weisbach : 


F2 
FI 

.10 

.20 

.30 

.40 

.50 

.60 

.70 

.80 

.90 

1.00 

k 

.362 

.338 

.308 

.267 

.221 

.164 

.105 

.053 

.015 

0 

84.  Loss  of  Head  Due  to  Obstruction  in  Pipe. — If  the  cross- 
section  of  a  pipe  at  any  point  is  partly  closed  by  an  obstruction, 
it  may  be  assumed  that  the  loss  of  head  is  due  mainly  to  the 


LOSS  OF  HEAD  CAUSED  BY  BEND.  75 

sudden  expansion  of  the  stream  just  beyond  the  obstruction, 
and  the  loss  may  be  estimated  as  in  the  case  of  a  sudden  enlarge- 
ment in  a  pipe  (Art.  82).  Thus,  let 

F  =  cross-section  of  pipe; 
-F'  =  area  at  obstructed  section; 
v  =  velocity  where  cross-section  is  F] 


The  formula  becomes 


2g  F' 


85.  Loss  of  Head  Caused  by  Bend.  —  The  loss  of  head  in  a 
curved  pipe  is  greater  than  that  in  an  equal  length  of  straight 
pipe.  No  rational  formula  can  be  given  for  estimating  its 
value,  and  the  empirical  rules  commonly  given  are  probably 
not  very  reliable.  It  is  reasonable  to  suppose  that  for  pipe  of 
a  given  size  the  loss  per  unit  length  is  greater  as  the  radius 
of  the  curve  is  less,  and  that  the  total  loss  due  to  a  curve  of 
given  radius  increases  with  the  total  angle  of  deflection,  i.e., 
with  the  length  of  the  curve.  For  curves  of  short  radius, 
having  the  same  total  angle  of  deflection  but  different  radii, 
the  usual  assumption  is  that  the  loss  of  head  increases  as  the 
ratio  of  the  radius  of  the  curve  to  that  of  the  pipe  decreases. 
For  larger  values  of  the  radius,  however,  the  total  loss  for  a 
curve  of  given  total  deflection  increases  with  the  radius  because 
of  the  increased  length  of  the  curve. 

For  90°  curves  of  short  radius,  the  formula  usually  given 
is  that  of  Weisbach,  who  assumed 


and  gave  for  k  the  empirical  formula 


in  which  r  is  the  radius  of  the  pipe  and  R  that  of  the  bend 


76 


EQUATION  OF  ENERGY  WITH  LOSSES. 


That  for  a  given  value  of  R  the  loss  of  head  should  increase 
with  the  diameter  of  the  pipe  seems  hardly  reasonable,  even 
when  R  is  small.  At  all  events,  Weisbach's  formula  should 
not  be  used  when  the  radius  of  the  curve  is  greater  than  four 
or  five  times  the  diameter  of  the  pipe. 

For  curves  of  longer  radius  experimental  data  are  too  lim- 
ited to  form  the  basis  of  any  general  formula.* 

86.  Hydraulic  Gradient. — If  piezometers  be  inserted  into  a 
pipe  at  various  points  of  its  length,  the  line  joining  the  highest 
points  of  all  the  piezometric  columns  is  called  the  hydraulic 
gradient.  In  Fig.  48,  A'Er  represents  the  hydraulic  gradient 
for  the  pipe  AB. 


Datum 
FIG.  48. 


The  general  nature  of  the  curve  of  the  hydraulic  gradient 
may  be  inferred  from  the  equation  of  energy.  Thue,  in  Fig. 
48,  let  zi,  Vi,  pi  refer  to  some  given  section  A,  and  2,  v,  p  to 
any  section  B,  down-stream  from  A.  Let  y  denote  the  eleva- 
tion above  datum  of  the  top  of  the  piezometric  column  at  any 
section,  y\  being  the  value  of  y  at  A.  The  equation  of  energy 

*  Experiments  on  the  loss  of  head  caused  by  90°  bends  in  pipes  of  12", 
16",  and  30"  diameter,  made  by  Gardner  S.  Williams,  Clarence  W.  Hubbell 
and  George  H.  Fenkell,  are  described  in  Vol.  XLVII  of  the  Transactions  of 
the  American  Society  of  Civil  Engineers.  The  loss  of  head  was  in  each  case 
measured  for  a  total  length  of  pipe  which  included  two  tangents  connected 
by  the  bend,  and  the  effect  of  the  bend  was  estimated  by  comparison  with 
the  loss  in  an  equal  total  length  of  straight  pipe.  Those  of  the  results  show- 
ing the  greatest  regularity  indicated  that  the  total  loss  of  head  due  to  each 
bend  in  the  30"  pipe  was  approximately  three  times  the  loss  caused  by  an 
equal  length  of  straight  pipe. 


HYDRAULIC  SLOPE.  77 

gives 

9±£       ,PI  ,  yi2    v2    w 

Z-\  --  =Z\-\  ---  HO  --  7;;  --  H-  i 

w  w     2g      2g 

if  Hr  is  the  head  lost  between  A  and  B.    Since  y  =  z  +  p/w,  the 
equation  may  be  written 


Consider  the  following  special  cases  : 

(1)  Suppose  the  cross-section  of  the  pipe  is  constant.    Then 
v  =  vi  and 

y-yi-H'. 

That  is,  the  hydraulic  gradient  falls,  between  any  two  sections, 
by  just  the  amount  of  the  head  lost  between  those  sections. 

On  the  assumption  of  no  loss  of  head,  the  hydraulic  gradient, 
in  case  of  uniform  cross-section,  would  be  a  horizontal  line. 

In  a  straight  pipe  of  uniform  interior  roughness  the  loss  of 
head  is  proportional  to  the  length,  and  hence  the  hydraulic 
gradient  has  a  uniform  downward  slope  in  the  direction  of  flow. 

A  bend  or  obstruction  would  cause  a  sudden  drop  in  the 
hydraulic  gradient. 

(2)  Suppose  the  cross-section  of  the  pipe  variable.     Then, 
in  addition  to  the  effect  of  loss  of  head,  the  hydraulic  gradient 
will  rise  or  fall  with  varying  velocity;    rising  as  the  velocity 
decreases  and  falling  as  it  increases. 

87.  Hydraulic  Slope.—  The  fall  of  the  hydraulic  gradient  per 
unit  length  of  the  pipe  is  called  the  hydraulic  slope. 

If  I  denotes  the  length  of  the  pipe  measured  from  a  fixed 
section  (as  A,  Fig.  48),  and  s  the  hydraulic  slope  at  the  point 
considered,  y  being  the  ordinate  of  the  hydraulic  gradient  at 
that  point  (as  B,  Fig.  48)  ,  we  have 

dy 


78  EQUATION   OF  ENERGY  WITH  LOSSES. 

If  the  hydraulic  slope  is  constant  from  A  to  J3, 


- 

If  the  cross-section  is  constant,  so  that  the  fall  of  the  hydrau- 
lic gradient  is  wholly  due  to  loss  of  head  (as  in  case  (1),  Art. 
86),  i/i  -?/=#'  and 

H' 

S=T' 

88.  Applications  of  Theory.  —  In  the  preceding  chapter  was 
treated  the  method  of  applying  the  general  equation  of  energy 
to  the  solution  of  problems  in  steady  flow,  neglecting  losses 
of  energy  by  dissipation.  The  general  method  there  outlined 
may  readily  be  extended  so  as  to  take  account  of  losses  of  head, 
the  value  of  the  term  Hf  in  the  equation  of  energy  being  expressed 
in  accordance  with  the  foregoing  principles.*  This  will  be 
illustrated  by  the  solution  of  an  example. 

EXAMPLES. 

1.  The  pipe  AB  (Fig.  49)  is  1000'  long  and  6"  in  diameter,  and  the 
length  AC  is  550'.  Compute  (a)  the  rafe  of  discharge;  (b)  the  pressure 
at  the  center  of  each  of  the  cross-sections  A,  B,  C. 


M    I 


Datum 
FIG.  49. 

Solution.— (a)  First  apply  the  equation  of  energy,  comparing  the 
points  M  and  N,  at  the  surfaces  of  the  two  reservoirs;  //i  denoting  the 

*  In  solving  the  following  examples,  the  coefficient  /,  for  estimating  the 
frictional  loss  of  head  in  pipes,  may  be  determined  by  Darcy's  formula  (Art. 
80). 


UNIVERSITY   I 


EXAMPLES.  79 

effective  head  at  M,  and  H2  that  at  Nt  while  Rf  denotes  the  total  head 
lost  from  M  to  N. 

Taking  datum  plane  as  shown  in  the  figure, 

#,=100,    tf2=60. 

The  value  of  H'  is  made  up  of  three  parts,  each  of  which  may  be  ex- 
pressed in  terms  of  the  velocity  of  flow  in  the  pipe.  Let  v  denote  this 
velocity;  then  we  have: 

(1)  The  loss  at  entrance  to  the  pipe  (Art.  81)  may  be  taken  as 


(2)  The  loss  between  A  and  B  due  to  friction  (Art.  80)  is 
I  v2 


Darcy's  formula  gives  /  =  .0232,  which  reduces  the  expression  for  the 
f  fictional  loss  to 

' 


(3)  The  loss  of  head  due  to  the  sudden  destruction  of  the  velocity  at 
the  point  of  discharge  into  the  reservoir  N  is  *(by  Art.  82,  putting 
F,/F»-Q) 


Combining  the  three  losses, 

#'=47.9-, 
V 

and  the  equation  #1  -H2  =  H'  becomes 


from  which 


100-60=47.9^, 


v2 

—  =0.833;  v  =  7.32  ft.  per  sec. 


The  rate  of  discharge  is 

g  =  Fv  =0.196X7.32  =  1.43  cu.  ft.  per  sec. 
(6)  Applying  the  energy  equation  to  sections  M  and  A   (the  latter 


80  EQUATION   OF  ENERGY  WITH   LOSSES. 

being  a  short  distance  within  the  pipe),    the  value  of  H'  is  Q.o(v*/2g) 
=0.42.     Also,  with  datum  as  before, 


arid  the  equation  Hi—H2  =H'  gives 

—  =  78.7  ft.  =  pressure  head  at  A. 

The  absolute  pressure  at  A  is  p2  +  p0,  since  in  this  solution  atmospheric 
pressure  has  been  called  zero. 

By  a  similar  method  the  pressure  head  at  C  is  found  to  be  46.3  ft. 

The  hydraulic  gradient  is  in  this  case  a  straight  line  except  near 
the  entrance  to  the  pipe.  Starting  at  the  surface  of  the  reservoir  M, 
it  falls  by  an  amount  l.5(vz/2g)  =  1.25  ft.  by  reason  of  the  entrance  loss 
and  the  velocity  head,  then  slopes  uniformly  by  reason  of  the  frictional 
loss,  falling  .03875  ft.  per  foot  of  length  of  the  pipe  until  the  surface  of 
the  reservoir  N  is  reached. 

2.  In  Fig.  49,  how  large  must  the  pipe  be  in  order  that  the  rate  of 
discharge  may  be  5  cu.  ft.  per  sec.?     In  that  case,  what  pressure  exists 
in  the  pipe  at  C? 

A  ns.  d  =  .82  ft.     Pressure  head  at  C  =46.4  ft.  above  atmosphere. 

3.  A  pipe  AB,  5000'  long,  is  to  discharge  5  cu.  ft.  per  sec.  in  the 
direction  AB.     The  point  B  is  higher  than  A,  and  it  is  required  that 
the  pressure  head  at  A  is  to  be  200',  and  that  at  B  100'.     What  must 
be  the  diameter  of  the  pipe?  Ans.  Nearly  1  foot. 

4.  A  pipe  AB  is  6000'  long  and  8"  in  diameter,  the  point  B  being  80' 
higher  than  A.     If  the  pressure  head  is  100'  at  B  and  190'  at  A,  in  which 
direction  is  the  flow,  and  what  quantity  is  discharged  per  sec.? 

Ans.  q  =  .623  cu.  ft.  per  sec.  in  direction  AB. 

5.  In  Ex.  4,  if  the  discharge  is  to  be  3  cu.  ft.  per  sec.  in  the  direction 
AB,  what  must  be  the  excess  of  pressure  at  A  over  that  at  5? 

Ans.  84.3  ft.  head. 

89.  Loss  of  Head  in  Capillary  Tubes.  —  Experiment  indicates 
that  the  loss  of  head  in  tubes  of  very  small  diameter  varies 
according  to  quite  different  laws  from  those  applying  to  ordinary 
pipes.  Poiseuille  gave  a  formula  equivalent  to  the  following, 
deduced  from  experiments  on  glass  tubes  from  .014  mm.  to 
.65  mm.  in  diameter: 

H'-k- 
H  -A. 


FLOW  OF  WELLS. 


81 


The  coefficient  k  depends  upon  the  viscosity  of  the  water,  and 
decreases  as  the  temperature  increases. 

90.  Flow    Through    Gravel. — When   water    flows     through 
porous  earth  or  gravel,  the  velocity  is  in  general  small,  and  the 
loss  of  head  is  found  to  vary  approximately  as  the  first  power 
of  the  velocity.    The  loss  per  unit  distance  along  the  stream 
depends  upon  the  character  of  the  material  through  which  the 
water  is  flowing,  being  greater  as  this  is  finer  and  more  compact. 

91.  Flow  of  Wells. — A  bed  of  gravel  or  other  porous  material 
lying  between  two  layers  of  clay  or  impervious  rock  may  carry 
water  under  pressure.    Such  a  case  is  represented  in  Fig.  50, 

A'  B' 


FIG.  50. 

which  shows  two  tubular  wells  A  and  B  which  have  been 
driven  from  the  surface  of  the  ground  down  to  and  through 
the  water-bearing  stratum  WW.  The  portion  of  each  tube 
within  the  porous  bed  is  perforated  so  as  to  permit  free  entrance 
of  water.  If  the  same  stratum  is  tapped  by  several  such  tubes, 
the  water  will  rise  in  all  to  the  same  level  if  the  water  in  the 
stratum  is  at  rest.  If,  however,  as  is  generally  the  case,  the 
water  is  flowing  through  the  porous  layer,  different  pressure 
columns  will  stand  at  different  levels. 


82  EQUATION  OF  ENERGY  WITH  LOSSES. 

Considering  a  case  in  which  the  water  is  at  rest,  let  the  two 
tubes  A,  B  (Fig.  50)  be  carried  high  enough  to  prevent  over- 
flow, and  let  A'E'  be  the  horizontal  plane  to  which  the  water 
rises.  Now  suppose  water  to  be  drawn  from  the  tube  A  at  a 
uniform  rate,  either  by  pumping  or  by  cutting  the  tube  off  at 
some  distance  below  A'  so  as  to  permit  overflow.  Flow  at 
once  begins  into  the  tube  and  toward  it  from  all  directions 
through  the  gravel,  and  a  steady  condition  ensues  in  which  the 
quantity  entering  the  tube  in  a  given  time  is  equal  to  the  quan- 
tity flowing  out.  In  this  steady  condition  there  is  a  definite 
relation  between  the  rate  of  discharge  and  the  drop  of  the 
water  surface  below  the  plane  A'E'.  The  nature  of  this  rela- 
tion depends  upon  the  way  in  which  loss  of  head  varies 
with  velocity  of  flow  both  in  the  porous  stratum  and  in  the 
tube. 

92.  Relation  between  Rate  of  Discharge  and  Fall  of  Water 
Surface. — Assuming  the  horizontal  plane  A'E'  as  datum,  let 
the  equation  Hi—Hz^H'  be  applied,   taking  the  up-stream 
point  at  a  great  distance  from  the  well  and  the  down-stream 
point  at  the  water  surface  in  the  tube. 
Let  v  =  velocity  of  flow  in  the  tube; 
q  =  rate  of  discharge ; 
u  =  drop  of  water  surface  =  A  A" . 

Then  #!=0,     H2=-u  +  ^-. 

To  express  Hf  in  terms  of  v  we  assume  the  law  stated  in 
Art.  90,  that  the  loss  of  head  due  to  flow  through  a  porous 
stratum  varies  directly  as  the  velocity,  other  conditions  remain- 
ing constant.  Although  the  velocity  has  different  values  at 
different  points,  the  values  at  all  points  may  be  assumed  to  vary 
directly  as  v,  so  that  the  total  loss  of  head  in  the  gravel  may 
be  expressed  by  a  term  Civ,  c\  being  a  constant.  The  loss  in 
the  tube  is  doubtless  ordinarily  small  in  comparison  with  that 
in  the  gravel,  but  to  take  account  of  this  and  the  loss  in  entering 
the  tube  a  term  cv2  may  be  introduced.  The  total  loss  of 


HYDRAULIC  GRADIENT  IN   CASE  OF  FLOWING  WELL.     83 
head  thus  takes  the  form 


and  the  equation  Hi—H2  =  H'  becomes 

v2 

U-— 

y 
from  which 


d  and  c2  being  constants.     Since  v  varies  directly  as  q, 


ki  and  k2  being  constants  whose  values  for  any  given  well  must 
be  determined  by  experiment. 
The  simpler  equation 


is  often  sufficiently  correct. 

These  results  are  verified  by  experiments  on  the  actual 
discharge  of  wells  for  different  heights  of  overflow. 

93.  Hydraulic  Gradient  in  Case  of  Flowing  Well.  —  Since 
the  hydraulic  gradient  falls  in  the  direction  of  flow  by  the 
amount  of  the  loss  of  head  (the  change  in  the  velocity  head 
being  in  this  case  inappreciable)  ,  the  gradient  is  a  surface  which 
rises  in  all  directions  in  going  from  the  well,  approaching  tan- 
gency  with  the  horizontal  surface  A'B1  '.  The  position  of  the 
water  surface  in  the  tube  B  indicates  the  height  of  the  hydraulic 
gradient  at  that  point.  The  difference  of  level  of  the  water 
surfaces  in  the  two  tubes  (corrected  slightly  for  the  velocity 
head  at  A"  and  the  loss  of  head  due  to  the  tube)  is  the  loss  in 
the  gravel  from  B  to  A.  The  less  the  porosity  of  the  gravel 
the  greater  is  the  value  of  this  loss,  i.e.,  the  rate  at  which  the 
hydraulic  gradient  rises  in  passing  away  from  the  well  is  greater 
as  the  stratum  is  less  pervious. 


84  EQUATION   OF   ENERGY  WITH   LOSSES. 

If  water  be  drawn  from  both  wells  at  the  same  time,  the 
drop  in  the  surface  of  each  will  be  due  in  part  to  its  own  dis- 
charge and  in  part  to  that  of  the  other  well.  The  quantity  of 
water  that  can  be  drawn  from  a  well  with  a  given  depression 
of  its  surface  below  the  static  level  A'B'  (Fig.  50)  may  thus 
be  influenced  in  an  important  manner  by  the  flow  of  other 
wells  in  the  vicinity. 


CHAPTER  VII. 

GENERAL  EQUATION  OF  ENERGY  WHEN  PUMP  OR  MOTOR 

IS  USED. 

94.  Equation  of  Energy  for  Stream  Flowing  Through  Motor. 

—  In  deducing  the  general  equation  of  energy  (Art.  60),  it  was 
assumed  that  no  mechanical  energy  is  imparted  to  the  water 
between  the  two  sections  A  and  B  (Fig.  25),  and  that  no  energy 
is  taken  from  it  except  such  as  is  dissipated  by  reason  of  fric- 
tion between  the  particles  of  water  and  the  pipe,  and  friction 
and  impact  among  the  particles  of  ^ 
water  themselves.  T  —  - 

If  a  motor  is  driven  by  the  stream, 
mechanical    energy  is    taken  by   the   j, 
motor  from  the  water,  and  this  must    j 
be  taken  into  account  in  forming  the 
equation  of  energy.    This  case  may  be  J  _ 
represented  by  Fig.  51,  in  which  M  is  FlG-  51- 

a  motor  through  which  water  flows  in  the  direction  AB. 

Let  Hf  now  mean  the  total  mechanical  energy  lost  by  the 
water  between  A  and  B,  per  pound  of  water  discharged,  and  let 


(1) 


in  which  h'  is  the  part  of  Hf  corresponding  to  energy  dissipated 
and  h"  the  part  corresponding  to  energy  received  by  the  motor. 
Then  the  energy  gained  by  the  volume  AB  per  unit  time  is 


W       2v 

85 


86          EQUATION  OF  ENERGY  WITH   PUMP  OR  MOTOR. 
and  the  energy  lost  is 


Equating, 


or 


"  =  Hl-H2-h'. 


(2) 


95.  Formula  for  Energy  Transferred  to  Motor. — The  amount 
of  energy  received  by  the  motor  from  the  stream  per  unit  time 
is 

and  the  horse-power  imparted  to  the  motor  is 


(4) 


W  being  in  pounds  per  second  and  HI,  H2,  h'  in  feet. 

Suppose  the  stream  passing 
through  the  motor-  flows  from 
one  reservoir  to  another,  the  sur- 
face of  the  discharge  reservoir 
being  h  ft.  lower  than  that  of  the 
supply  reservoir  (Fig.  52) .  Tak- 
ing sections  A  and  B  at  the  reser- 
voir surfaces,  we  have 


- 


FIG.  52. 


Hence  in  this  case  we  may  write 


(5) 


Wh'  being  the  energy  dissipated  per  second  in  the  entire  stream 
from  A  to  B.  The  effect  of  the  frictional  losses  of  energy  is 
thus  to  decrease  by  Wh'  the  energy  imparted  to  the  motor  per 
second,  which  amounts  practically  to  decreasing  the  available 
fall  of  the  water  by  h'. 


STREAM  FLOWING  THROUGH   PUMP.  87 

If  the  motor  discharges  directly  into  the  air,  equations  (3) 
and  (4)  still  apply,  but  instead  of  reducing  this  to  the  form  (5) , 
it  is  more  convenient  to  change  the  notation.  If  h  now  denotes 
the  fall  from  surface  of  supply  reservoir  to  point  of  outflow 
from  motor,  and  v-2  the  velocity  of  outflow, 


and  (3)  becomes 
V     .     :.-•        wv.w^-V-*?) (6) 

96.  Equation  of  Energy  for  Stream  Flowing  Through  Pump. 
If  the  flow  of  a  stream  is  maintained  or  aided  by  a  pump, 
the  mechanical  energy  given  up  by 

the  pump  to  the  water  must  be  taken  ^          — p r 

into  account  in  forming  the  equation         /f 
of    energy.     Let  this  case  be   repre- 
sented   by    Fig.    53,    P   representing 
the  pump,  and  AB  being  the  direction 
of  flow.  '  '       L  \ 

Let  ft'"  denote  the  energy  received 


by   the   water    from    the   pump    per  FIG.  53. 

pound  of  water  discharged,  and  let  hf  denote  as  above  the 
energy  lost  by  dissipation  between  A  and  B  per  pound  of  the 
discharge.  Then  H'  being  the  net  loss  of  head  from  A  to  B, 

H'  =  h'-hm  ........     (7) 

Reasoning  as  in  Art.  60,  we  may  write  at  once 


from  which  ft'"=#2-#i+ft'.     ......     (8) 

It  is  seen  that  the  action  of  the  pump  causes  the  effective 
head  to  increase  in  the  direction  of  flow  by  the  amount  h'"  '. 
If  h'"  is  greater  than  h'  ',  H2  is  greater  than  Hi,  i.e.,  the  total 
loss  of  head  Hr  is  negative. 


88 


EQUATION  OF  ENERGY  WITH  PUMP  OR  MOTOR. 


97.  Formula  for  Energy  Used  in  Pumping.— The  total  me- 
chanical energy  supplied  by  the  pump  per  second  is 


=  W(H2-Hi+k'). 


(9) 


Here  h'  must  be  understood  to  include  energy  lost  by  hydraulic 
friction  within  the  pump  as  well  as  in  other  parts  of  the  stream. 
The  rate  of  working  of  the  pump,  in  horse-power,  is 


H.P. 


Wk"> 
"550" 


W 


I.    ...     .     (10) 


Suppose  water  is  pumped  from  one  reservoir  to  another 
(Fig.  54),  the  vertical  lift  between  reservoir  surfaces  being  h. 
B        Taking  the  sections  A  and  B  at  the  two 
surfaces,  we  have 


and  (9)  may  be  written  in  the  form 


The  work  done  in  pumping  is  thus 
FIG.  54.  .  Till  •"'»••'« 

the  same  as  would  be  done   in    lifting 

the  water  a  height  k  +  k'  against  gravity  if  all  waste  of  energy 
could  be  avoided. 

EXAMPLES. 

1.  Water  is  supplied  to  the  motor  M  (Fig.  55)  from  the  head-race  A 
and  conducted  from  M  to  the  tail-race  D.     The  supply-pipe  and  waste- 
pipe  are  each  6"  in  diameter.     Suppose  that  no  energy  is 
lost  by  dissipation  in  any  part  of  the  apparatus,  and  that 
the  discharge  is  3  cu.  ft.  per  sec.    Compute  the  pressure 
at  B  and  at  C,  and  the  H.P.  transmitted  to  the  motor. 

Ans.  Pressure  head  at  B=  p0/w  +  6.4';  pressure 
head  at  C=p,/w- 17.6';   H.P.  =  15. 


14' 


p[ 


2.  Water  is  pumped  from  one  reservoif  into  another 
through  1000'  of  6"  pipe.  The  surface  of  the  second 
reservoir  is  20'  higher  than  that  of  the  first.  If  the 
quantity  delivered  is  3  cu.  ft.  per  sec.,  (a)  at  what  rate 
is  energy  imparted  to  the  water  by  the  pump?  (6)  What 
fraction  of  this  energy  is  utilized  in  lifting  the  water?  (c)  The  losses 


Cp 
C 


10' 


T 


FIG.  55. 


EXAMPLES. 


89 


of  energy  in  the  flow  are  equivalent  to  what  added  lift?      (d)  Draw 
hydraulic  gradient. 

Arcs,  (a)  66.8  H.P.     (6)  About  10%.     (c)  176  ft. 

3.  Water  flows  from  one  reservoir  into  another,  the  surface  of  the 
second  being  10'  lower  than  that  of  the  first.     The  flow  takes  place 
through  a  pipe  8"  in  diameter  and  1600'  long,  the  intake  end  projecting 
into  the  reservoir,  while  the  other  end  discharges  below  the  water  surface, 
(a)  If  the  flow  is  due  to  gravity  alone,  what  quantity  is  discharged  per 
second?     (6)  In  order  to  double  the  rate  of  discharge,  at  what  rate  must 
energy  be  given  to  the  water  by  the  pump? 

Ans.  (a)  1.19  cu.  ft.  per  sec.     (6)  8.12  H.P. 

4.  In  Fig.  56,  C  is  the  cylinder  of  a  piston-pump  which  is  lifting 
water  from  the  reservoir  B  and  discharging  it  at  D.    The  diameter  of 
the  pipe  is  everywhere  2",  and  the  radius 

of  the  bend  is  3". 

(a)  At  what  maximum  rate  can  water 
be  pumped  without  causing  a  pressure 
head  of  less  than  4'  (absolute)  at  any 
point  in  the  pipe?  Where  will  the  mini- 
mum pressure  occur? 

(6)  Compare  the  pressures  just  below 
and  just  above  the  pump  cylinder, 
(c)  Compute  the  H.P.  of  the  pump. 

[Estimate  the  loss  due  to  a  bend  by  formula  of  Weisbach.    Solve, 
also,  disregarding  this  loss  and  see  whether  it  is  important.] 

Ans.  (a)  0.257  cu.  ft.  per  sec.  (6)  Below,  p/w=Q  (absolute); 
above,  p/w  =35.8'  above  atmosphere,  (c)  2.06  H.P.  The  pressure 
head  above  the  cylinder  is  increased  0.4'  by  the  bend,  and  the 
H.P.  is  increased  about  0.5%. 


, 50- 


Di 


FIG.  56. 


CHAPTER  VIII. 
FLOW  IN  PIPES:   SPECIAL  CASES. 

98.  Flow  From  Reservoir  Through  Pipe.— If  a  pipe  leads  from 
a  reservoir  and  discharges  into  the  atmosphere,  formulas  for 
the  velocity  of  flow  and  the  rate  of  discharge  may  be  obtained 
by  applying  the  general  equation  of  energy  Hi-H2  =  H'. 

Referring  to  Fig.  57,  let 


FIG.  57. 

h  =  total  fall  from  reservoir  to  center  of  outflowing  stream; 
v  =  mean  velocity  in  pipe  ; 
i/=    "  of  outflowing  stream; 

/  =  length  of  pipe; 
d  =  diameter  of  pipe. 

Writing  the  equation  of  energy  for  the  stream  AB,  taking 
datum  plane  through  the  center  of  the  stream  at  B, 

Hi=h  =  effective  head  at  A; 

v'2 
#2  =  o~  =  effective  head  at  B't 


90 


FLOW   FROM  RESERVOIR  THROUGH   PIPE.  91 

If  the  pipe  is  of  uniform  section,  straight  and  unobstructed, 
Hf  is  the  sum  of  two  parts, — the  loss  at  entrance,  and  the  loss 
by  friction  in  the  pipe.  If  there  are  bends  or  obstructions  of 
any  kind,  or  expansions  or  contractions  of  the  cross-section, 
these  cause  additional  losses  which  may  be  taken  as  approxi- 
mately proportional  to  the  square  of  the  velocity. 

CASE  I.  Let  the  cross-section  of  the  outflowing  stream  or 
jet  be  equal  to  that  of  the  pipe,  so  that  i/  =  v;  then 


The  losses  of  head  may  be  expressed  as  follows : 

v2 
The  entrance  loss  is  ra^-,  where  w  =  l  for  projecting  and 

0.5  for  non-projecting  pipe  (Art.  81). 

I  v2 
The  frictional  loss  in  the  pipe  is  /  -y-  5-,  in  which  /  has  the 

meaning  explained  in  Art.  80. 

The  loss  due  to  bends  or  obstructions  is  expressed  by  a 

v2 
term  n~-,  the  value  of  n  depending  upon  the  circumstances  of 

each  particular  case. 
Combining  all  losses, 


The  equation  Hi~H2=:H/  now  becomes 


The  method  of  solving  this  equation  in  particular  cases  will 
be  illustrated  below. 

CASE  II.  Let  the  cross-section  of  the  jet  be  less  than  that 
of  the  pipe. 

If  F  =  cross-section  of  pipe  and  F'  =  that  of  jet,  let  F=cF'. 
Then  v'  =  cv  and 

(cv)2 
Hi-Hz-h-^-. 


92  FLOW   IN  PIPES:    SPECIAL  CASES. 

To  the  losses  of  head  occurring  in  Case  I  must  be  added  a 
loss  due  to  the  contraction  of  the  stream  at  the  point  of  out- 

v'2 
flow.    This  loss  may  be  expressed  by  a  term  k  ~-,  the  value  of 

k  depending  upon  the  ratio  of  contraction  and  the  construction 
of  the  pipe  and  orifice.    Combining  all  losses, 


and  the  equation  of  energy  becomes 

(cv)2     /  fl\v2 


.     .. 
2g  '  d/2g          2g 

Equation  (1)  is  a  special  case  of  (2),  in  which  c  =  l, 
Equation  (2)  may  be  written  in  the  torm 


(2) 


-  .....     (3) 


From  it  may  be  computed  any  one  of  the  quantities  v,  h,  d,  if 
the  others  are  known.  For  certain  applications  it  is  conve- 
nient to  introduce  the  rate  of  discharge  q  instead  of  v.  Thus, 
suppose  it  is  required  to  estimate  the  size  of  the  pipe  which 
will  give  a  certain  rate  of  discharge,  the  total  fall  and  length 
of  pipe  being  known. 

Since  v  =  q/F  -=4q/nd2,  equation  (3)  may  be  written 


from  which 

d*  =  ^-h([(l+k)c2+m+n]d+fi),   ....     (4) 

or  d5  =  Ad  +  B,      .......    (4r) 

A  and  B  being  constants. 


FLOW   FROM  RESERVOIR  THROUGH   PIPE.  93 

In  many  cases,  especially  for  long  pipes  discharging  as  in 
Case  I,  the  term  Ad  is  small  in  comparison  with  B,  and  the 
solution  may  be  made  by  successive  approximations. 

As  an  example,  let  h  =  5Q',  £  =  2500',  g  =  10  cu.  ft.  per  sec., 
c  =  l;  &  =  0,  m  =  l,  n  =  Q.  These  values  reduce  equation  (4)  to 


For  a  first  approximate  solution  assume  /=.02  and  neglect 
the  term  2d,  which  is  small  in  comparison  with  2500/.  This 
gives  <i  =  1.20,  and  the  value  of  /  given  by  Darcy's  formula  is 
.0213.  Using  these  results  in  the  above  equation, 

d5  =  .  0504(2.40  +  53.25)  -2.81, 
d  =  1.23'  =  nearly  15". 

If  the  term  Ad  is  not  small  in  comparison  with  B,  solution 
by  trial  is  less  easy,  but  is  always  possible.   ; 

EXAMPLES, 

1.  Let  the  diameter  of  the  pipe  be  6",  the  length  850',  and  the  fall 
to  the  point  of  outflow  40'.     Assume  that  the  pipe  projects  into  the 
reservoir,  and  that  the  discharge  takes  place  through  a  nozzle  2"  in 
diameter,  causing  a  loss  of  head  0.1(-y'2/2<7),  vf  being  the  velocity  of  the 
jet.     (a)  Compute    the    rate    of    discharge,     (b)  Draw    the    hydraulic 
gradient.  Ans.  (a)  0.88  cu.  ft.  per  sec, 

2.  In  Ex.  1,  let  the  diameter  of  the  pipe  be  12",  the  remaining  data 
being  unchanged.     Compute  q,  and  draw  the  hydraulic  gradient. 

Ans.  9  =  1,05  cu.  ft.  per  sec. 

3    In  Ex.  1,  compute  q  on  the  assumption  that  the  cross-section  of 
the  jet  equals  that  of  the  pipe.     Draw  the  hydraulic  gradient. 

Ans,  9  =  1.55  cu.  ft.  per  sec. 

4.  Let  h  =40',  /=850',  q  =  16  cu.  ft.  per  sec,     Assuming  no  contrac- 
tion of  the  stream  at  outlet,  and  taking  the  coefficient  of  entrance  loss 
as  0.5,  compute  the  size  of  the  pipe.     Draw  the  hydraulic  gradient. 

Ans.  Diameter  =  about  15". 

5.  Water  is  conducted  from  a  reservoir  to  a  point  300'  lower  than 
the  reservoir  surface.     The  conduit  is  a  4"  pipe,  1200'  long,  projecting 
into  the  reservoir.     Discharge  occurs  into  the  atmosphere  through  a 
nozzle,  delivering  a  stream  1"  in  diameter,  the  coefficient  of  velocity  for 


94  FLOW  IN   PIPES:    SPECIAL  CASES. 

the  nozzle  being  0.9.  (a)  Compute  the  rate  of  discharge.  (6)  Compute 
the  available  energy  of  the  jet  per  second,  (c)  If  the  jet  drives  a  motor 
with  an  efficiency  of  U.75,  wnat  H.P.  is  realized?  (d)  What  fraction  of 
the  total  energy  due  to  the  fail  is  utilized? 

[The  loss  of  head  in  the  nozzle  may  be  expressed  in  terms  of  the 
velocity  of  the  jet  and  the  coefficient  of  velocity  in  the  same  way  as  in 
the  case  of  a  short  tube  (Art.  78).] 

Ans.  (a)  0.602  cu.  ft.  per  sec.     (c)  9.7  H.P.     (d)  63%.  . 

6.  Taking  data  as  in  example  5,  and  assuming  the  supply  of  water 
to  be  unlimited,  what  sized  nozzle  will  deliver  the  greatest  amount  of 
energy  to  the  motor  in  a  given  time?  [Assume  the  coefficient  of  velocity 
of  the  nozzle  to  be  0.9  for  all  cases.] 

Ans.  Diam.  of  nozzle  =0.287  Xdiam.  of  pipe. 

99.  Pressure  at  a  Given  Point  to  Have  an  Assigned  Value.— 
Let  it  be  required  to  conduct  water  from  a  reservoir  to  a  certain 
A    _/  _  point,  the  rate  of  discharge  being 

[  given,  and  the  pressure  within 
/  the  pipe  at  the  point  of  delivery 


having  a  specified  value.     Thus, 
in  Fig.  58,  let  B  be  a  point  at 
FlG-  58>  which  water  is  to  be  delivered, 

the  pressure  head  there  being  required  to  have  the  value  y. 

Let  /  =  length  of  pipe  to  B,  d  =  diameter,  /i*=  depth  of  B 
below  reservoir  surface. 

Taking  datum  plane  through  the  center  of  the  pipe  at  B, 
the  values  of  the  effective  head  at  A  and  B  are 

v2 
Hi=h,    H.2  =  y+7r; 

while  the  loss  of  head  between  A  and  B  is 

l\  v2 


The  equation  of  energy  therefore  becomes 

v2     I  l\  v2 


This  is  identical  with  equation  (1)  with  h  —  y  substituted  for  h. 


BRANCHING  PIPE. 


95 


EXAMPLE. 

Water  is  to  be  delivered  through  a  pipe  1800'  long  to  a  point  70' 
below  the  level  of  the  reservoir  surface.  The  rate  of  discharge  is  to  be 
15  cu.  ft.  per  sec.,  and  the  pressure  head  at  the  point  of  delivery  40'. 
What  should  be  the  size  of  the  pipe?  Ans.  d  =  18". 

100.  Branching  Pipe. — If  water  is  delivered  through  a  main 
pipe  with  branches,  the  equation  of  energy  must  be  applied 
to  each  portion  of  the  pipe  separately,  sufficient  data  being 
known  or  assumed  to  make  the  problem  of  flow  determinate. 

Thus,  as  a  simple  case,  consider  the  main  pipe  AB  with 
two  branches  BC,  BD  (Fig.  59),  and  let  the  given  data  be  the 


FIG.  59. 

rate  of  discharge  at  C  and  at  D,  and  the  pressure  at  each  of 
the  three  points  B,  C,  D.  The  method  of  treatment  may  be 
illustrated  by  a  numerical  case. 

Let  8  cu.  ft.  per  sec.  be  delivered  at  C  and  10  cu.  ft.  per 
sec.  at  D;.  let  the  depth  of  B  below  the  reservoir  surface  be  100', 
that  of  C  80',  and  that  of  D  110';  let  the  length  of  the  main 
pipe  to  B  be  2000',  that  of  BC  1200',  and  that  of  BD  1600'; 
and  let  the  pressure  head  have  the  value  60'  at  B,  30'  at  C, 
and  50'  at  D. 

Let  d,  d',  d"  be  the  diameters  of  the  three  pipes  AB,  BC, 
CD;  and  v,  vf,  v"  the  corresponding  velocities  of  flow. 

The  losses  of  head  at  entrance  to  the  branch  pipes  cannot 
be  accurately  estimated,  but  will  be  small  in  comparison  with 
the  frictional  losses  in  the  pipes  and  will  be  neglected. 

For  the  pipe  AB  the  equation  is 


40         _ 
~ 


2000X1* 
d    >> 


96  FLOW  IN  PIPES:    SPECIAL  CASES. 

1  8 
from  which,  since  v=     2>  the  va^ue  °f  ^  may  be  found  as 


in  Art.  98. 

For  the  pipe  BC  we  have 


'2 


12001/2 


v2  v'2        1200/2 

hence  77-  +  10  —  ^—  =  /  —  -rr-  —  . 

2g  2g  d'    2g 

The  value  of  d  having  been  previously  determined,  v  is  known. 

o 

And  since  i/  =     »2/:,  d'  may  be  determined  by  solving  the 

last  equation  in  the  usual  way.     The  branch  BD  may  be  treated 
in  the  same  manner. 

EXAMPLE. 

Complete  the  solution  of  the  above  numerical  case. 

101.  Relation  of  Pipe  to  Hydraulic  Gradient.  —  For  a  uni- 
form pipe,  straight  and  unobstructed,  the  hydraulic  gradient 
(Art.  86)  is  uniform  for  the  entire  length.  If  the  stream  dis- 
charges into  the  air  without  contraction,  the  pressure  at  the 
outlet  end  is  atmospheric,  while  at  a  short  distance  within  the 
intake  end  it  is  less  than  atmospheric  by  a  small  amount,  equiv- 
alent to  the  velocity  head  plus  the  entrance  loss.  Between 
these  points  the  hydraulic  gradient  is  a  straight  line,  as  shown 
in  Fig.  57.  Gradual  curves  in  the  pipe  line  will  not  materially 
affect  the  uniformity  of  the  hydraulic  slope,  but  the  velocity 
which  corresponds  to  a  given  total  fall  will  be  less  when  there 
are  bends  than  in  a  straight  pipe  of  the  same  length. 

In  laying  a  long  pipe  line,  both  horizontal  and  vertical  curves 
must  often  be  introduced  because  of  the  contour  of  the  ground. 
In  planning  the  line  it  is  of  especial  importance  to  see  that 


EFFECT  OF  AIR  IN  CHECKING  FLOW.  97 

the  pipe  shall  everywhere  lie  below  the  hydraulic  gradient  as 
it  will  exist  when  the  rate  of  discharge  has  its  greatest  value. 

Fig.  60  shows  a  case  in  which  a  large  part  of  the  pipe  is 
above  the  hydraulic  gradient  AB.  Assuming  full  flow  to  exist, 
the  pressure  in  the  pipe  everywhere  between  C  and  B  is  less 
than  atmospheric.  The  resultant  pressure  upon  the  pipe  is 
thus  from  without,  but  (aside  from  the  danger  of  collapsing  the 
pipe)  the  decrease  of  pressure  will  not  of  itself  interfere  with 
the  flow  unless  the  pipe  goes  high  enough  to  reduce  the  pressure 
to  absolute  zero.  Practically,  however,  it  would  be  difficult  to 
start  full  flow  under  such  conditions,  and  even  if  started  it 
would  not  continue  unless  the  pipe  were  absolutely  air-tight 


FIG.  60. 

and  the  water  free  from  air.  Since  air  is  always  carried  by 
natural  waters,  there  would  soon  be  an  accumulation  of  air  in 
the  higher  parts  of  the  pipe  which  would  check  the  flow  even 
if  there  were  no  leakage  from  without. 

102.  Effect  of  Air  in  Checking  Flow. — The  collection  of  air 
at  a  summit  may  seriously  interfere  with  the  discharge  of  a 
pipe  even  when  everywhere  below  hydraulic  gradient,  but  so 
long  as  the  pressure  is  above  that  of  the  atmosphere  the  air 
may  be  removed  by  means  of  a  valve  provided  for  the  purpose. 

To  understand  the  action  of  air  in  checking  the  flow,  con- 
sider the  case  represented  in  Fig.  61.  Here  the  hydraulic  gra- 
dient for  full  flow  is  everywhere  above  the  pipe.  But  if  air 
collects  in  the  bend  C,  the  stream  will  finally  be  divided  into 
two  parts,  the  surface  beyond  C  being  forced  down  toward  7), 


98 


FLOW  IN  PIPES:    SPECIAL  CASES. 


as  shown  at  Y.  The  flow  will  be  retarded,  but  will  not  be 
stopped  unless  the  pressure  of  the  confined  air  becomes  great 
enough  to  balance  the  static  pressure  due  to  the  water  column 
AX  plus  atmospheric  pressure.  If  D  is  far  enough  below  B, 
the  condition  represented  in  Fig.  62  will  finally  ensue  and  flow 
will  cease.  The  pressure  of  the  confined  air  will  then  just  bal- 
ance each  of  the  two  equal  *  columns  AX,  YB. 

A 


FIG.  61. 

The  effect  of  air  upon  the  hydraulic  gradient  is  shown  in 
Figs.  61  and  62.  With  full  flow  the  gradient  would  be  a  line 
of  nearly  uniform  slope  from  A  to  B.  As  air  accumulates,  the 
pressure  increases  in  the  up-stream  part  of  the  pipe  and  de- 
creases in  the  down-stream  part,  piezometer  columns  standing 


FIG.  62. 

at  equal  heights  above  the  surfaces  X  and  Y  as  shown  in  Fig. 
61.  When  the  flow  is  stopped,  as  in  Fig.  62,  the  gradient  con- 
sists of  two  horizontal  lines,  one  lying  in  the  plane  of  the 
reservoir  surface,  the  other  in  a  horizontal  plane  as  far  above 
Y  as  X  is  below  the  reservoir  surface. 

*  This  neglects  the  weight  of  the  confined  air,  by  reason  of  which  the 
pressure  at  Y  would  be  slightly  greater  than  that  at  X. 


CHEZY'S  FORMULA.  99 

103.  Long  Pipe  with  Full  Discharge.  —  In  case  of  a  long  pipe 

I  v2 
the  friction  loss,  expressed  by  the  term  /  -7  75-,  is  very  great  in 

comparison  with  the  other  losses  of  head,  and  also  in  compari- 
son with  the  velocity  head.  Equation  (1)  therefore  reduces 
practically  to  the  form 


f 

=fd2g' 


and  equation  (5)  to  the  form 


In  the  majority  of  practical  problems  relating  to  the  design  of 
water  supply  systems  these  simplified  equations  are  used,  the 
value  of  the  neglected  terms  being  within  the  limits  of  relia- 
bility of  the  data. 

104.  Che"zy's  Formula.  —  The  formula  for  loss  of  head  in  a 
uniform  straight  pipe,  given  in  Arts.  79  and  80,  is  often  written 
in  another  form.  Introducing  the  hydraulic  radius  (the  ratio 
of  the  area  of  the  cross-section  to  its  circumference)  instead  of 
the  diameter,  the  formula 


I    V2 

becomes  F/== 


Solving  for  v,  writing  s  for  -r,  and  introducing  a  new  coefficient 
c  such  that 


we  have  v  =  c\/rs, 

which  is  known  as  Chezy's  formula.* 

*  This  is  the  formula  commonly  used  in  the  discussion  of  flow  in  open 
channels.     See  Chapter  XI.  9 


100  FLOW  IN   PIPES:    SPECIAL  CASES. 

If  there  are  bends  in  the  pipe,  or  if  the  roughness  of  the 
surface  varies  in  different  parts,  the  losses  of  head  in  different 
equal  lengths  are  unequal,  and  the  hydraulic  slope  s  varies 
along  the  pipe.  This  is  usually  the  ease  in  practice.  In  such 
cases  the  formula  v  =  cVrs  is  still  often  employed,  s  being 
regarded  as  the  average  hydraulic  slope  for  the  entire  length, 
i.e.,  s  =  Hf  /I,  where  Hf  is  the  total  loss  of  head  in  the  length  /. 
The  value  of  c  must  be  taken  less  than  in  the  case  of  straight 
pipe. 

The  formula  for  /  given  in  Art.  80,  based  upon  Darcy's 
results,  is  equivalent  to  the  following  formula  for  c: 


Further  discussion  of  the  values  of  /  and  c  is  given  in  Chap- 
ter IX. 

The  two  formulas 


are,  as  above  shown,  identical.  Although  the  latter  has  been 
much  employed,  the  former  is  preferable  for  the  reason  that  / 
is  an  abstract  number,  while  c  depends  upon  the  system  of  units 
employed. 

In  solving  the  following  examples,  let  Chezy's  formula  be 
employed,  the  value  of  c  being  determined  from  the  formula 
above  given. 


EXAMPLES.  101 

EXAMPLES. 

1.  A  pipe  18"  in  diameter  will  discharge  what  quantity  of  water  per 
second  if  the  hydraulic  gradient  falls  10'  in  a  length  of  SOOO'? 

Ans.  </=4.22  cu.  ft.  per  sec. 

2.  A  pipe  8500'  long  is  to  discharge  40  cu.  ft.  per  sec.,  with  a  total 
fall  of  25'  in  the  hydraulic  gradient.     What  should  be  the  diameter? 

Ans.  About  3.1'. 

3.  If  the  pipe  in  Ex.  2  is  designed  so  as  to  just  satisfy  the  conditions 
stated,  how  much  will  the  hydraulic  gradient  fall  if  the  rate  of  discharge 
becomes  50  cu.  ft.  per  sec.?  Ans.  About  39'. 

4.  If  two  equal  pipes  are  to  be  substituted  for  the  one  pipe  in  Ex.  2, 
what  diameter  should  they  have?  Ans.  About  28". 


CHAPTER  IX. 
FRICTIONAL  LOSS  OF  HEAD  IN  PIPES. 

105.  Importance  of  Frictional  Loss. — The  so-called  "  friction 
head,"  or  frictional  loss  of  head  in  pipes,  is  the  most  important 
factor  affecting  the  discharging  capacity  of  pipe  lines  or  dis- 
tribution systems  in  practice,  since  other  losses  are  commonly 
small  in  comparison  with  it.    The  present  chapter  will  there- 
fore be  devoted  to  a  fuller  discussion  of  methods  of  determining 
or  estimating  this  loss. 

106.  Method  of  Measuring  Loss  of  Head. — The  measure- 
ments which  must  be  made  in  order  to  determine  the  actual 
loss  of  head  in  a  given  length  of  pipe  in  which  a  condition  of 
steady  flow  exists  have  been  indicated  in  Art.  73.     The  effec- 
tive head  at  any  section  involves  the  three  quantities  elevation, 
pressure,  and  velocity  of  flow.     Of  these  the  last  is  often  the 
most  difficult  to  measure  with  the  requisite  accuracy.     Methods 
of  accomplishing  it  are  considered  briefly  in  Chapter  XIII.    If 
the  cross-sectional  areas  of  the  pipe  at  the   two  sections  at 
which  the  effective  head  is  to  be  measured  are  equal,  the  veloc- 
ity terms  disappear  from  the  value  of  the  lost  head;   but  even 
in  this  case  the  velocity  corresponding  to  any  measured  loss  of 
head  must  be  known  in  order  that  the  results  may  have  value 
as  a  guide  to  design. 

The  measurement  of  the  difference  in  elevation  of  the  two 
cross-sections,  with  the  accuracy  requisite  in  experimental 
work  of  this  kind,  may  involve  considerable  labor,  especially 
in  the  case  of  long  pipe  lines  laid  under  the  conditions  of  actual 

practice.    The  necessity  of  making  such  measurements  may  be 

102 


METHOD  OF  MEASURING  LOSS  OF  HEAD. 


103 


obviated  if  it  is  possible  to  shut  off  the  flow  and  read  the  pres- 
sure gauges  when  hydrostatic  conditions  exist  throughout  the 
pipe.  Thus,  let  AB  (Fig.  63)  be  the  length  of  pipe  in  which 


J|_JL_L 


FIG.  63. 

loss  of  head  is  to  be  determined,  and  suppose  piezometer  tubes 
to  be  connected  at  A  and  B.  Let  the  position  of  the  top  of 
each  piezometer  column  be  read  upon  a  fixed  vertical  scale, 
2/i  and  y2  being  simultaneous  values  of  the  two  readings.  Let 
YI,  YZ  be  values  of  y\,  2/2  when  the  flow  is  cut  off. 

When  flow  takes  place  the  loss  of  head  in  the  entire  line 
above  A  is 


while  the  loss  in  the  entire  line  above  B  is 


Hence  the  loss  from  A  to  B  is 


If  Vl=V2, 


Thus  the  actual  heights  of  the  piezometer  columns  above  a 
common  datum  need  not  be  measured,  the  above  value  of  Hf 
being  independent  of  the  positions  of  the  zero  points  of  the 
two  fixed  scales.* 

*  If  the  zero  points  are  so  located  that  Yi  <  Y9,  a  constant  may  be  added 
to  all  values  of  yl  such  that  Yl  —  Y2  will  be  positive;  every  value  of  yl  —  yz 
will  then  be  positive. 


104  FRICTIONAL  LOSS  OF  HEAD  IN  PIPES. 

The  same  method  is  applicable  if  any  other  kind  of  pres- 
sure gauge  be  used  instead  of  a  simple  water  piezometer,  and 
the  above  formula  for  Hf  still  holds  if  y\  and  y2  denote  simul- 
taneous readings  of  the  two  gauges,  and  FI,  Y2  the  values  of 
2/i,  2/2  under  static  conditions;  it  being  understood  that  all 
gauge  readings  are  in  feet  of  water.  If  the  gauges  are  other- 
wise graduated,  either  the  readings  or  the  final  value  of  H' 
may  be  reduced  to  equivalent  water  column  by  applying  the 
proper  factor.* 

107.  Formulas  for  Friction  Loss. — It  appears  not  to  be  pos- 
sible to  express  the  frictional  loss  of  head  in  pipes  accurately,  in 
any  simple  way,  in  terms  of  the  several  quantities  upon  which 
its  value  depends.  For  straight  pipes  of  a  given  kind  the  loss 
in  a  given  length  varies  chiefly  with  diameter  and  velocity  of 
flow.  Many  attempts  have  been  made  to  establish  a  formula 
expressing  its  value  in  terms  of  these  two  variables,  which 
should  be  applicable  to  pipes  of  a  given  kind  throughout  the 
entire  range  of  diameters  and  velocities  likely  to  be  met  in 
practice,  and  which  could  be  applied  to  different  kinds  of  pipe 
by  a  proper  choice  of  constants.  Hitherto  such  attempts  have 
not  been  successful,  and  there  is  little  reason  to  expect  future 
success  in  this  direction.  The  most  careful  experiments  made 
with  pipes  under  practical  conditions  show  that  the  so-called 
friction  loss  is  materially  influenced  by  factors  which  cannot 
be  controlled  by  the  experimenter,  and  which  in  the  practical 
use  of  the  pipes  are  sure  to  vary.  One  of  these  factors  is  tem- 
perature. Another  is  the  character  of  the  pipe  surface,  which 
may  vary  from  month  to  month,  or  even  from  day  to  day,f 
enough  to  appreciably  change  the  loss  of  head  due  to  a  given 

*  If  a  mercury  manometer  of  the  kind  described  in  Art.  75  is  employed, 
the  fluctuation  of  the  surface  of  the  mercury  in  the  reservoir  must  be  taken 
into  account  in  reducing  gauge-reading  to  equivalent  water  column.  Also,  in 
case  the  pressure  is  so  great  that  the  column  of  mercury  is  of  considerable 
length,  it  may  be  necessary  to  apply  a  correction  for  changes  of  temperature, 
for  which  purpose  the  length  of  the  column  must  be  known  approximately. 
See  Trans.  Am.  Soc.  C.  E.,  Vol.  XL,  p.  481;  Vol.  XLIV,  p.  39. 

t  Darcy,  Recherches  expe"rimentales,  p.  107. 


BASIS  OF  FORMULAS  FOR  FRICTION   LOSS. 


105 


velocity.  In  view  of  these  facts  it  seems  futile  to  seek  a  single 
general  formula  by  which  the  friction  loss  in  any  proposed  pipe 
line  can  be  predicted  with  great  accuracy. 

Most  of  the  formulas  that  have  been  proposed  have  a  certain 
basis  of  theory.  This  theory,  and  some  of  the  more  important 
of  the  formulas,  will  be  considered  in  the  following  articles. 

108.  Theoretical    Basis    of   Formulas  for   Friction    Loss. — 

Consider  a  straight  pipe  whose  cross-section  is  uniform  but  of 
any  shape.  Let  F  denote  the  area  and  C  the  perimeter  of 
the  cross-section,  and  let  r  =  F/C  (as  in  Art.  79).  Let  A  and 
B  (Fig.  64)  be  two  cross-sections  whose  distance  apart  is  I,  and 


FIG.  64. 

let  pi,  p2  denote  the  pressures  at  centroids  of  sections  A  and  B 
respectively.  Conceive  piezometer  tubes  communicating  with 
the  pipe  at  the  two  sections,  and  let  t/i,  y2  denote  elevations  of 
the  two  water  columns  above  a  chosen  datum  plane;  z\,  22  being 
heights  of  centroids  of  the  two  sections  above  the  same  datum. 

The  body  of  water  between  A  and  B  being  in  a  condition 
of  steady  flow,  the  external  forces  acting  upon  this  body  are  in 
equilibrium.  Let  these  be  resolved  in  the  direction  of  the  flow. 

The  system  of  external  forces  consists  of  the  normal  pres- 
sures exerted  by  the  adjacent  water  upon  the  two  cross-sectional 
areas  A  and  5,  the  weight  of  the  body  of  water  AB,  and  the 
normal  and  tangential  forces  exerted  by  the  pipe  surface,  the 
tangential  or  frictional  forces  being  opposite  in  direction  to  the 


106  FRICTIONAL  LOSS  OF  HEAD  IN   PIPES. 

flow.    Let  P  denote  the  sum  of  the  frictional  forces  on  the 
entire  area  of  contact  of  the  body  A B  with  the  pipe. 

Since  in  any  cross-section  of  the  stream  the  pressure  varies 
according  to  the  hydrostatic  law,  the  total  pressure  on  the 
cross-section  is  equal  to  the  product  of  the  area  into  the  inten- 
sity of  pressure  at  the  centroid  (Art.  17).  Hence 

piF  =  total  pressure  on  cross-section  A', 
p2F  =    "          "        "  "  B. 

The  weight  of  the  body  of  water  between  A  and  B  is  wFl,  and 
its  component  in  the  direction  of  the  flow  is 

wFl  cos  <t>  =  wF(zi  -  z2) . 

Equating  to  0  the  sum  of  the  resolved  forces  in  the  direction 
of  the  flow, 

piF-p2F+wF(zi-z2)-P=Q. 

Since  pi/w  +  Zi  =  yi  and  p2/w  +  z2  =  y2,  and  since  y\-y2  =  H'  = 
loss  of  head  between  A  and  B,  the  equation  may  be  written 


Thus  the  loss  of  head  is  expressed  in  terms  of  the  total  frictional 
force  exerted  by  the  pipe  surface  upon  the  water. 
If  the  frictional  force  per  unit  area  is  P',  so  that 

P  =  CIP', 
we  may  write 

OF  =  j_  P_ 

wF      r    w' 

r  being  the  hydraulic  radius  as  already  defined  (Art.  79). 

Experiment  indicates  that  the  value  of  P'  is  independent  of 
the  pressure,  but  varies  with  the  velocity  of  the  relative  motion. 
If  the  law  of  variation  were  known,  the  above  formula  would 


MEAN   VELOCITY    IN   PLACE  OF  SURFACE  VELOCITY.     107 

give  the  loss  of  head  in  terms  of  the  surface  velocity.    Thus,  if 
vi  is  the  surface  velocity,  and  if 


the  formula  becomes 


109.  Introduction  of  Mean  Velocity  in  Place  of  Surface 
Velocity.  —  Equation  (1)  would  be  of  little  use  even  if  the  form 
of  the  function  <£(i?i)  were  known,  since  the  quantity  of  impor- 
tance is  not  the  surface  velocity  v\,  but  the  mean  .velocity  v. 

If  it  be  assumed  that  there  is  a  fixed  relation  between  v\ 
and  v  which  is  independent  of  the  size  of  the  pipe,  there  results 
an  equation  of  the  form 

.    ,:   .--'••   <    .    .    (2) 


If  this  assumption  were  correct,  the  dependence  of  Hf  upon 
the  size  of  the  pipe  would  be  expressed  by  a  very  simple  rela- 
tion. Thus,  since  for  circular  pipes  r  =  d/4,  loss  of  head  would 
vary  inversely  as  diameter.  The  fact  that  this  simple  law  is 
not  verified  by  experiment  indicates  that  the  relation  between 
mean  velocity  and  surface  velocity  is  not  independent  of  the 
size  of  the  pipe.  Some  of  the  earlier  writers  on  Hydraulics, 
however,  assumed  that  mean  velocity  varies  directly  as  surface 
velocity,  and  thus  deduced  the  form  of  the  function  F(v)  in 
equation  (2)  from  the  experimental  law  governing  the  friction 
between  a  solid  and  a  fluid. 

110.  —  Application  of  Experimental  Law  of  Fluid  Friction.  — 

The  frictional  force  per  unit  area  exerted  by  a  solid  body  upon 
a  fluid  flowing  uniformly  over  its  surface  is  found  (a)  to  be 
independent  of  the  pressure  and  (b)  to  vary  approximately  as 
the  square  of  the  velocity  of  flow,  except  for  quite  small  velocities, 


108  FRICTIONAL  LOSS  OF  HEAD  IN   PIPES. 

and  (c)  to  vary  nearly  in  direct  ratio  with  the  velocity  of  flow 
if  this  is  very  small. 

Applying  this  to  the  pipe  problem,  the  value  of  Pf  may  be 
expressed  approximately  in  the  following  form: 


(3) 


Assuming  that  v\  varies  directly  as  vt  equation  (2)  may  be 
written 

H'  =  —(av+bv2)  .......     (4) 

t 

111.  Formula  of  Proqy.  —  Equation  (4)  is  the  formula  given 
by  Prony,  who,  from  such  experimental  data  as  were  available, 
deduced  numerical  values  of  a  and  b  which  were  supposed  to 
hold  for  all  sizes  of  pipe.  It  was  supposed,  also,  that  the  char- 
acter of  the  surface  tif  the  pipe  had  little  influence  upon  the 
values  of  these  coefficients. 


Formula  of  Darcy.  —  Equation  (4)  was  also  employed 
by  Darcy,  who  found,  however,  that  the  coefficients  a  and  b 
vary  greatly  with  the  character  of  the  pipe  surface,*  and  also 
that  they  are  not  independent  of  the  diameter.  From  his 
experiments  he  deduced  empirical  formulas  of  the  forms 


B 


(5) 


giving  numerical  values  for  a,  /?,  a:',  /?',  applicable  to  pipes  of 
about  the  degree  of  smoothness  of  new  cast  iron. 

Darcy's  experiments,  as  well  as  many  made  subsequently, 
indicate  not  only  that  the  term  av  in  equation  (4)  is  unimpor- 
tant in  comparison  with  bv2  except  for  quite  small  velocities, 
but  also  that  its  importance  is  less  the  rougher  the  surface  of 

*  Being,  for  example,  about  twice  as  great  for  cast-iron  pipes  fouled  by 
some  years  of  use  as  for  the  same  pipes  new  or  thoroughly  cleaned. 


CHEZY'S  FORMULA.     KUTTER'S  FORMULA.  109 

the  pipe.  For  pipes  in  practical  use,  and  for  the  range  of 
velocities  ordinarily  existing,  Darcy  recommended  as  sufficiently 
correct  a  formula  of  the  form 

fl"-7*i*      .......     (6) 

And  for  values  of  d  within  the  range  of  his  experiments  (from 
0.5  inch  to  20  inches  approximately)  he  gave  the  following  for- 
mula for  the  coefficient  61  : 


The  equivalent  of  this  formula,  with  numerical  values  of  the 

coefficients,  is  given  in  Art.  80. 

/ 

113.  ChSzy's  Formula.  —  It  will  be  seen  that  equation  (6)  is 
really  the  same  as  the  formula  already  given  in  two  forms  (Arts. 
80,  104)  : 


v  =  cVrs.       .;,'...     .  •  .     (9) 

In  the  latter  form  it  is  often  known  as  Chezy's  formula,  and 
is  identical  with  the  formula  commonly  employed  in  estimat- 
ing the  discharge  of  streams  flowing  in  open  channels.* 

114.  Kutter's  Formula.—  Kutter's  formula  is  an  empirical 
expression  for  the  value  of  c  in  terms  of  r,  s,  and  a  certain  quan- 
tity n  called  the  coefficient  of  roughness,  whose  value  depends 
upon  the  character  of  the  surface  over  which  the  water  is  flow- 
ing. The  relation  is  as  follows: 

I™  +41.654^ 


r  being  in  feet  and  v  in  feet  per  second. 

*  See  Art.  128. 


110  FfUCTlOiNAL  LOSS  OF  HEAD  IN   PIPES. 

Although  this  complex  formula  was  based  upon  experi- 
mental data  for  flow  in  open  channels,  it  has  often  been  applied 
to  pipes  and  closed  conduits,  especially  those  of  large  size.  Its 
use  cannot,  however,  be  recommended,  except  in  the  absence 
of  experimental  knowledge  directly  applicable  to  the  case  in 
hand. 

Some  indication  of  the  values  of  n  to  be  used  for  pipes  of 
different  kinds  will  be  given  below. 

115.  Exponential  Formula.—  The  foregoing  discussion  has 
brought  out  the  fact  that  the  loss  of  head  in  a  given  length  of 
pipe  varies  approximately,  but  not  exactly,  as  the  square  of 
the  velocity  of  flow  in  pipes  of  the  same  size,  and  inversely  as 
the  diameter  in  pipes  of  different  sizes;  all  being  assumed  alike 
in  the  character  of  the  interior  surface.  It  was  found  by 
Osborne  Reynolds,*  in  experiments  with  lead  tubes  of  one- 
fourth  inch  and  one-half  inch  diameter,  that  the  loss  of  head 
varied  as  a  power  of  the  velocity  whose  index  was  less  than  2. 
A  similar  relation  was  observed  by  Lampe  in  experiments  with 
a  pipe  about  16.5  inches  in  diameter.  From  a  study  of  the 
experimental  results  of  Darcy  and  Poiseuille,  Reynolds  found 
a  similar  exponential  relation,  but  with  the  index  varying  with 
the  character  of  the  pipe.  A  study  of  other  experimental 
data  obtained  with  both  small  and  large  pipes  shows  that  the 
variation  of  loss  of  head  with  velocity  in  a  given  pipe  may  often 
be  closely  represented  by  the  formula 


with  constant  values  of  A  and  m.  The  value  of  m  is  usually 
less  than  2,  but  varies  considerably  for  different  pipes,  and  in 
some  cases  has  been  found  to  be  greater  than  2.  There  is 
some  evidence  tending  to  show  that  for  a  given  character  of 
pipe  surface  m  is  independent  of  the  diameter,  and  that  it 
increases  with  the  roughness  of  the  surface.  Thus  a  value  of 
1.72  was  found  by  Reynolds  for  the  small  lead  tubes,  and 
about  the  same  value  has  been  found  for  other  quite  smooth 
*  Philosophical  Transactions  of  the  Royal  Society,  1883,  Part  III,  p.  975. 


EXPONENTIAL  FORMULA.  Ill 

pipes,*  while  for  pipes  quite  rough  by  incrustation  or  other- 
wise a  value  of  about  2  has  been  found.! 

Whether  the  character  of  the  pipe  surface  is  the  only  im- 
portant factor  affecting  the  value  of  the  exponent  cannot  be 
determined  from  known  experimental  data,  but  it  seems  likely 
that  for  pipes  laid  under  the  conditions  of  practice  it  may  vary 
with  the  curvature  of  the  pipe  line,  the  temperature  of  the 
water,  and  perhaps  other  factors. 

The  value  of  A  probably  depends  upon  the  character  of  the 
pipe  surface,  as  well  as  upon  the  diameter.  A  series  of  experi- 
ments upon  pipes  alike  in  all  respects  except  in  size  would 
doubtless  show  a  regular  variation  of  A  with  the  diameter.  It 
has  been  supposed  that  this  also  may  prove  to  be  an  exponen- 
tial relation,  but  for  large  pipes  experimental  data  for  the 
establishment  of  such  a  law  are  almost  wholly  lacking.  Such 
evidence  as  there  is  seems  to  show  that  if  A  varies  with  a  power 
of  d,  the  exponent  will  be  negative  and  numerically  greater  than 
unity,  t 

It  was  pointed  out  by  Reynolds  that  the  above  formula  is 
a  convenient  one  for  representing  experimental  data  because  of 
the  ease  with  which  the  constants  can  be  determined  graphic- 
ally. Taking  logarithms,  the  formula  becomes 

log  Hf  =  log  A+m  log  v. 

If  a  series  of  experimental  values  of  H'  and  v  satisfies  this  for- 
mula, the  locus  whose  coordinates  are  log  Hf  and  log  v  will  be  a 
straight  line.  The  slope  of  this  line  determines  the  value  of 
m,  and  its  intercept  on  the  axis  along  which  log  Hf  is  meas- 
ured gives  the  value  of  log;!.  The  plotting  is  facilitated  by 
the  use  of  logarithmic  cross-section  paper.  The  plotting  of  the 
results  in  this  manner  shows  at  once  whether  the  assumed  form 
of  formula  agrees  well  with  the  experimental  data,  and  also 
gives  a  simple  determination  of  the.  constants  in  case  it  does. 

*  See,  for  example,  Trans.  Am.  Soc.  C.  E.,  Vol.  LI,  p.  52. 

t  Phil.  Trans.  Roy,  Soc.,  1883,  Part  111,  p.  981.  Trans.  Am.  Soc.  C.  E. 
Vol.  XXXV,  p.  258. 

J  Tables  for  practical  use,  based  upon  an  exponential  formula,  have 
recently  been  published  by  Gardner  S.  Williams  and  Allen  Hazen. 


112  FRICTIONAL  LOSS  OF  HEAD   IN   PIPES. 

116.  Critical  Velocity.— The  experiments  of  Reynolds  showed 
that  there  was  a  certain  velocity  below  which  the  loss  of  head 
varied  nearly  as  the  first  power  of  the  velocity,  and  above 
which  it  varied  as  a  higher  power.    The  value  at  which  the  law 
changes  he  called  the  critical  velocity.    By  observations  of  the 
flow  in  glass  tubes  he  found  that  for  very  low  velocities  the 
flow  took  place  quite  accurately  in  parallel  filaments,  while 
above  the  critical  velocity  the  stream  became  turbid. 

In  practical  Hydraulics  velocities  below  the  critical  value 
rarely  or  never  need  be  considered.  It  is  to  the  range  of  veloc- 
ities above  the  critical  point  that  the  above  discussion  of  the 
exponential  formula  refers. 

117.  Formula  Adopted.— For    the    purpose    of    expressing 
working  rules  for  estimating  loss  of  head,  the  equivalent  for- 
mulas 


v  =  cv  rs, 

will  here  be  employed.  For  convenience  values  both  of  the 
friction  factor  /  and  of  the  coefficient  c  will  in  most  cases  be 
given.  When  one  of  these  quantities  is  known  the  other  can 
be  computed  from  the  relation 


As  already  pointed  out,  /  is  an  abstract  number  and  therefore 
independent  of  the  system  of  units  employed,  while  c  depends 
upon  the  units  of  length  and  time,  c2  being  of  the  same  dimen- 
sions as  g. 

Owing  to  the  imperfection  of  the  theory  upon  which  the 
above  formulas  are  based,  /  and  c  cannot  be  regarded  as  con- 
stants for  a  given  kind  of  pipe,  but  vary  with  both  diameter 
and  velocity. 

118.  Values  of  Friction  Coefficients. — Many  attempts  have 
been  made,  by  study  of  known  experimental  data,  to  establish 


OF  THfe 

UNIVERSITY 

OF 

VALUES  OF  FRICTION 


definite  values  of  the  friction  factors  for  the  kinds  of  pipe  used 
in  practical  hydraulic  works.  Formulas  or  tables  professing  to 
give  such  values  with  great  accuracy  cannot,  however,  be 
accepted  with  confidence,  because  of  the  great  uncertainties 
in  the  reliability  of  the  data  upon  which  they  are  necessarily 
based. 

The  experiments  of  Darcy  *  are  probably  the  most  trust- 
worthy series  yet  made  covering  any  considerable  range  of 
diameters  and  velocities.  Upon  the  results  he  based  two  for- 
mulas, one  expressing  as  accurately  as  possible  the  variation 
of  the  friction  factor  with  both  diameter  and  velocity,  the 
other  neglecting  the  variation  with  velocity.  The  latter,  which 
was  recommended  by  its  author  for  practical  use,  was  equivalent 
to  the  following,  in  which  d  is  the  diameter  in  feet  : 


This  is  for  pipes  of  new  or  clean  cast  iron,  or  other  pipes  of 
equal  smoothness.  For  pipes  tuberculated  or  otherwise  rough- 
ened by  some  years  of  use  Darcy  recommended  that  the  values 
of  /  given  by  this  formula  be  doubled. 

The  pipes  used  in  Darcy  's  experiments  ranged  in  diameter 
from  0.0122  meter  to  0.5  meter.  The  most  reliable  experiments 
made  since  by  others  indicate  that  his  formula  gives  safe  values, 
not  only  within  this  range,  but  for  larger  sizes,  and  the  formula 
has  often  been  used  for  diameters  as  great  as  4  feet.  Table 
III  (page  114),  based  upon  the  formula,  is  carried  to  about 
double  the  largest  size  experimented  upon  by  Darcy. 

There  is  some  evidence  that  the  coefficients  in  column  3 
of  the  table  are  somewhat  too  great  for  pipes  as  smooth 
as  ordinary  clean  cast-iron  water  pipes.  It  is,  however,  impor- 
tant in  engineering  design  to  allow  a  margin  of  safety  to  cover 
both  the  large  element  of  uncertainty  in  existing  knowledge 
regarding  friction  losses  in  pipes  of  any  given  character,  and 
the  uncertainty  in  any  given  case  regarding  the  actual  present 

*  Recherches  experimentales  relatives  au  mouvement  de  1'eau  dans  les 
tuyeaux.  Henry  Darcy.  Paris,  1857. 


114 


FRICTIONAL   LOSS  OF  HEAD  IN    PIPES. 


and  future  character  of  the  particular  pipe  under  consideration. 
The  tabulated  coefficients,  both  for  smooth  and  for  rough  pipes, 
probably  err  on  the  side  of  safety,  unless  it  be  for  the  case  of 
pipes  which  have  suffered  exceptional  deterioration  with  long 


use. 


TABLE  III. 


VALUES  OF  THE  FRICTION  FACTOR  /  AND  OF  THE  COEFFICIENT  c,  BASED  UPON 
THE  FORMULA  OF  DAR  Y. 


1 

2 

3 

4 

5 

6 

Diameter. 

Smooth  Pipe 

Rough  Pipe 

Inches. 

Feet. 

/ 

c 

/ 

c 

1 

.0833 

.0398 

80 

.0796 

57 

2 

.167 

.0298 

93 

.0596 

66 

3 

.250 

.0265 

99 

.0530 

70 

4 

.333 

.0248 

102 

.0496 

72 

6 

.500 

.0232 

105 

.0464 

74 

8 

.667 

.0224 

107 

.0448 

76 

10 

.833 

.0219 

108 

.0438 

77 

12 

1.000 

.0216 

109 

.0432 

77 

14 

1.167 

.0213 

110 

.0426 

78 

16 

1.333 

.0211 

110 

.0422 

78 

18 

1.500 

.0210 

111 

.0420 

78 

24 

2.000 

.0207 

111 

.0417 

79 

30 

2.500 

.0206 

112 

.0412 

79 

36 

3.000 

.0205 

112 

.0410 

79 

The  degree  of  roughness  or  smoothness  of  any  given  pipe 
must  be  a  matter  for  the  judgment  of  the  engineer.  But,  except 
in  the  rare  cases  in  which  pipes  are  laid  for  temporary  use  only, 
the  design  must  be  governed  by  probable  future  conditions 
rather  than  by  the  character  of  the  pipe  when  new. 

119.  Large  Pipes. — Although  there  is  little  experimental 
evidence  regarding  the  variation  of  /  and  c  with  the  diameter 
in  the  case  of  large  pipes,  this  variation  appears  to  be  unim- 
portant. As  regards  variation  with  the  velocity,  there  is  evi- 
dence that,  except  for  quite  small  velocities,  this  is  of  little 
practical  importance  even  with  smooth  pipes,  and  is  quite 
inappreciable  in  the  case  of  rough  pipes,  such  as  those  of  riveted 
steel  or  almost  any  pipe  after  long  use.  For  pipes  from  3  ft. 


FRICTION   FACTORS  FOR  LARGE  PIPES. 


115 


to  6  ft.  in  diameter  it  is  therefore  sufficient  to  use  values  of  / 
and  c  which  are  independent  of  velocity  and  diameter,  varying 
only  with  the  character  of  the  pipe. 

The  general  range  of  these  values  is  indicated  in  the  follow- 
ing table.  It  is  of  course  to  be  understood  that  actual  pipes 
present  all  gradations  of  roughness,  and  the  four  cases  specified 
in  the  table  can  only  serve  as  a  very  general  and  approximate 
guide.  The  values  tabulated  (except  for  Case  I)  are  designed 
to  allow  for  the  resistance  due  to  such  curves  as  are  likely  to 
exist  in  practical  cases. 

TABLE  IV. 

FRICTION  FACTORS  AND  COEFFICIENTS  FOR  LARGE  PIPES. 
(Diameter  from  3  ft.  to  6  ft.) 


Case 

Typical  Pipe. 

/ 

c 

n 

I. 

(Very  smooth.) 

Exceptionally  smooth  cast-iron  pipe, 
new  or  thoroughly  cleaned,  without 
bends  

013 

140 

Oil 

II. 

(Smooth  ) 

Ordinary  new  cast-iron  or  wooden 
pipe  with  some  bends  

018 

120 

0125 

III. 

(Rough.) 

New  riveted  pipe,  with  some  bends,  or 
any  pipe  after  some  years  of  use.  .  . 

.023 

106 

.014 

IV. 

Any  pipe  after  long  use  

046 

75 

019 

(Very  rough.) 

120.  Kutter's  Coefficient  of  Roughness.—  Kutter's  formula 
(Art.  114)  has  come  into  quite  general  use  as  a  practical  guide 
in  estimating  the  discharging  capacity  of  large  pipes.  This 
formula  assumes  to  take  account  of  the  roughness  of  the  sur- 
face by  the  value  assigned  to  the  coefficient  n.  It  also  makes 
c  vary  with  diameter  and  velocity.  The  values  of  n  given  in 
the  last  column  of  the  above  table  are  about  the  average  values 
implied  by  the  corresponding  values  given  for  c. 


CHAPTER  X. 

EQUATION  OF  ENERGY  FOR  STREAM  OF  LARGE  CROSS- 

SECTION. 


Streams  of  Large  Cross-section.  —  The  theoretical  dis- 
cussion given  in  Chapter  IV,  leading  to  the  general  equation  of 
energy  for  a  steady  stream,  involved  the  assumption  that  the 
intensity  of  pressure  may  be  regarded  as  uniform  throughout 
any  cross-section,  and  that  all  particles  passing  the  section  have 
equal  velocities.  These  assumptions  cannot  be  supposed  to  be 
even  approximately  true  when  the  cross-section  is  large,  as  in 
the  case  of  rivers  and  canals.  Let  us  consider  whether  the 
theory  requires  important  modification  when  variations  of 
pressure  and  of  velocity  in  a  cross-section  are  taken  into  account. 


Variation   of  Pressure   Throughout  a  Cross-section.  — 

If  all  particles  move  in  straight  lines  parallel  to  the  axis  of  the 
stream,  the  pressure  in  any  normal  cross-section  varies  with 
the  depth  according  to  the  hydrostatic  law.  For  let  A  and  B 
be  two  points  such  that  AB  is  perpendicular  to  the  direction 
of  flow,  and  consider  a  prism  of  water  of  small  cross-section 
whose  axis  is  AB.  Resolving  all  forces  acting  upon  this  prism 
in  the  direction  AB,  the  sum  of  such  resolved  forces  must  be 
zero,  since  there  is  no  acceleration  in  this  direction.  Hence 
the  reasoning  of  Art.  11  may  be  applied,  leading  to  the  same 
conclusion. 

123.  Variation  of  Velocity  in  a  Cross-section.  —  The  way  in 
which  the  velocity  varies  throughout  the  cross-section  has  been 
the  subject  of  considerable  investigation,  both  experimental 
and  theoretical.  The  following  discussion  is,  however,  inde- 
pendent of  any  assumption  as  to  the  actual  distribution  of 

116 


ENERGY   PASSING  A  CROSS-SECTION.  117 

velocities ;  though,  as  will  be  seen,  a  certain  term  in  the  equation 
of  energy  cannot  be  evaluated  unless  the  law  of  distribution 
is  known. 

124.  Energy  Passing  a  Cross-section. — To  determine  the 
energy  passing  any  cross-section  per  unit  time,  we  may  reason 
substantially  as  in  Art.  59,  introducing  such  changes  as  are 


necessitated  by  the  variation  of  pressure  and  of  velocity  through- 
out the  section.     Referring  to  the  cross-section  A  (Fig.  65),  let 

v  =  velocity  at  any  point  in  the  section; 

F  =  whole  area  of  the  section; 

g= rate  of  discharge  across  area  F; 

vf  =  -p  =  mean  velocity ; 

p  =  pressure  at  point  whose  velocity  is  #; 
z  =  height  of  that  point  above  datum. 

Energy  transferred  across  the  section  by  pressure. — The  energy 
transferred  across  an  element  dF  of  the  cross-section  by  reason 
of  the  pressure  and  velocity  may  be  computed  as  in  Art.  59. 
Its  value  per  second  is  vpdF,  and  therefore  the  energy  thus 
transferred  across  the  entire  section  per  second  is 


fpvdF, 


(1) 


the  integration  covering  the  whole  section  F. 

Potential  energy  carried  across  the  section. — The   potential 
energy  carried  across  an  elementary  area  dF  per  second  is 

wzv  dF, 


118  EQUATION   OF  ENERGY:  LARGE  STREAM. 

and  the  potential  energy  passing  the  whole  area  F  per  second  is 

yfzvdF.      .     .    .    .    .    .    *     (2) 


w 


Sum  of  potential  energy  and  energy  transferred  by  pressure.  — 
The  sum  of  the  values  (1)  and  (2)  may  be  written 


w 


Now  since  the  pressure  varies  throughout  the  section  according 
to  the  hydrostatic  law,  in  passing  to  different  points  in  the 
section  p/w  increases  (or  decreases)  by  just  the  amount  of 
decrease  (or  increase)  of  2;  that  is,  z  +  p/w  is  constant  through- 
out the  section.  Denoting  this  sum  by  y,  it  is  seen  that  y  is 
equal  to  the  height  above  datum  of  the  top  of  a  piezometer 
column  communicating  with  the  pipe  at  some  point  of  the 
given  cross-section. 

Since  y  is  constant  in  the  integration,  the  value  of  the  in- 
tegral becomes 


(3) 


Kinetic  energy  passing  the  section.  —  The  amount  of  kinetic 
energy  passing  the  elementary  area  dF  in  one  second  is 


hence  the  amount  passing  the  whole  section  F  is 


w 


This  integral  cannot  be  evaluated  unless  the  variation  of  v 
throughout  the  section  is  known.  It  may,  however,  be  shown 
that  it  is  greater  than  the  value  which  would  be  obtained  if 


ENERGY   PASSING  A  CROSS-SECTION.  119 

the  mean  velocity  ?/  were  assumed  to  apply  throughout  the 
section.     That  is, 


Thus,  let  v  =  v'+u;   then 


fu  dF  + 
But 


since  it  denotes  the  excess  of  actual  velocity  at  any  point  over 
mean  velocity.     We  may  therefore  write 


The  last  integral  is  positive;  for 


and  since  vf  -\-u  or  v  is  always  positive  even  when  u  is  negative, 
it  follows  that  v'H--^  must  be  positive  for  all  values  of  u.    The 

o 

inequality  (5)  is  thus  proved. 

The  total  kinetic  energy  carried  across  the  section  per  unit 
time  may  therefore  be  written 


K  being  a  positive  quantity  equal  to  the  last  term  of  (6). 

Total  energy  parsing  the  section.  —  The  total  energy  passing 


120  EQUATION  OF  ENERGY:  LARGE  STREAM. 

the  section  in  one  second  is  the  sum  of  the  values  (3)  and  (7). 
It  may  be  written 


in  which  k=K/wq. 

The  value  of  the  energy  passing  any  cross-section  per  unit 
weight  of  water  discharged  may  therefore  be  expressed  in  either 
of  the  following  forms: 

V2  V       V2 


in  which  v  is  now  written  for  the  mean  velocity  in  the  cross- 
section.  As  shown  above,  z  and  p  may  refer  to  any  point  in 
the  section,  their  sum  being  constant. 

Comparing  this  result  with  that  reached  in  Art.  59  it  is 
seen  that  the  effect  of  the  variation  of  velocity  is  represented 
by  the  term  k. 

125.  Equation  of  Energy  in  Case  of  Large  Stream.  —  The 
reasoning  of  Art.  60  may  be  applied  to  the  case  of  a  large 
stream,  using  the  expression  above  deduced  for  energy  passing 
a  section.  With  suffixes  (i)  and  (2)  to  refer  to  up-stream  and 
down-stream  sections  respectively,  we  have 


in  which  the  values  of  z  and  p  refer  to  any  point  of  the  section. 
Instead  of  z-\  p/w  we  may  write  y;  y  being  constant  for  each 
section,  and  meaning  the  height  above  datum  of  the  top  of  a 
piezometer  column  communicating  with  the  pipe  at  any  point 
of  the  perimeter  of  the  section. 

As  shown  above,  k\  and  k%  are  positive  quantities,  the  value 
of  each  depending  upon  the  distribution  of  velocities  in  the 
corresponding  section.  The  quantity  k  cannot  be  zero  unless 
the  velocity  has  the  same  value  throughout  the  section,  and 
will  be  great  in  proportion  as  the  variation  of  velocity  is  great 


GENERAL  EQUATION  OF  ENERGY.         121 

If  the  two  sections  compared  have  the  same  size  and  shape,  it 
may  reasonably  be  assumed  that  ki=k-2\  in  this  case  the  equa- 
tion of  energy  takes  the  same  form  as  if  the  velocity  were 
uniform  throughout  every  cross-section.  For  sections  of  un- 
equal size  and  consequent  unequal  mean  velocities  no  such 
assumption  can  rationally  be  made. 

Ordinarily,  for  circular  pipes,  no  great  error  results  from 
using  the  equation  of  energy  in  the  ordinary  form,  as  above  in 
the  discussion  of  small  pipes.  This  statement  is  justified  by 
experiment;  and  it  may  be  concluded  either  that  the  value  of 
k  is  usually  small,  or  that  its  values  for  sections  of  different  size 
are  nearly  equal. 


CHAPTER  XI. 
UNIFORM  FLOW  IN  OPEN  CHANNELS. 

126.  General  Principles.  —  The  principles  employed  in  the 
discussion  of  flow  in  pipes  apply  in  the  main  to  streams  in 
open  channels.  The  chief  difference  arises  from  the  fact  that 
in  the  case  of  a  stream  in  an  open  channel  the  upper  surface 
is  in  contact  with  the  atmosphere.  From  this  it  follows  (a)  that 
the  pressure  at  this  surface  has  a  known  constant  value,  and 
(&)  that  the  frictional  resistance  to  flow  at  this  surface  is  much 
less  than  at  the  surface  of  the  channel  itself. 

The  way  in  which  the  velocity  of  flow  varies  throughout 
any  cross-section  is  in  general  unknown,  and  can  be  determined 
only  by  experiment.  The  effect  of  this  variation  upon  the 
general  equation  of  energy  for  steady  flow  has  been  considered 
in  Chapter  X,  the  discussion  there  given  applying  to  open  as 
well  as  to  confined  streams.  In  the  following  discussion  it  is 
usually  the  mean  velocity  in  a  cross-section,  rather  than  the 
actual  velocities  in  different  parts  of  the  section,  that  will  be 
considered.  If  this  mean  velocity  is  denoted  by  v,  the  relation 


=  constant  for  all  cross-sections 

holds  for  an  open  channel,  if  the  flow  is  steady. 

The  following  discussion  will  be  restricted  to  the  case  of 
steady  flow  in  a  channel  of  uniform  cross-section  and  slope. 
In  the  cases  of  most  importance  it  will  further  be  true  that 
the  depth  of  the  water  is  the  same  at  all  sections,  so  that  the 
water  cross-sections  are  all  alike  in  size  and  shape.  The  flow 

122 


FORMULA   FOR  MEAN  VELOCITY.  123 

is  then  said  to  be  uniform.    The  problem  of  non-uniform  flow 
will  be  considered  in  the  next  chapter. 

127.  Uniform  Flow. — Consider  the  case  of  a  channel  of  uni- 
form cross-section  and  slope  and  of  considerable  length,  into 
the  upper  end  of  which  water  is  admitted  at  a  constant  rate. 
In  a  short  time  the  flow  will  become  steady,  and  it  will  also 
become  practically  uniform  throughout  the  greater  part  of  the 
straight  and  uniform  channel  under  consideration.     When  this 
condition  is  reached,  the  velocity  of  flow  will  depend  upon 
(a)  the  slope  of  the  channel,  (b)  the  character  of  the  surface 
over  which  the  water  flows,  and  (c)  the  dimensions  of  the  cross- 
section. 

(a)  The  velocity  is  greater  as  the  slope  is  greater. 

(b)  The  velocity  is  greater  as  the  channel  surface  is  smoother. 

(c)  The  greater  the  length  of  the  wetted  perimeter  of  the 
cross-section  in  comparison  with  its  area,  the  greater  the  re- 
tarding effect  of  friction,  and  therefore  the  less  the  velocity. 

128.  Formula  for  Mean  Velocity  in  Case  of  Uniform  Flow. 

— No  theoretical  discussion  of  flow  in  open  channels,  even  in 

the  simplest  case  of  uniform  flow,  can  serve  as  more  than  a 

rough  guide  in  the  solution  of  practical  problems.     The  formula 

most  commonly  employed  in  practice  is  that  of  Chezy,  already 

given  as  applying  to  pipes.     The  reasoning  of  Arts.  108-110 

may  be  applied  with  slight  change  to  the  case  of  an  open  channel. 

Let  Fig.  66  represent  a  longitudinal  section  of  the  stream, 

and  consider  the  body  of  water  between  two  cross-sections  at 

A 


FIG.  66. 


A  and  B.    The  flow  being  steady,  this  body  has  no  acceleration, 
and  the  forces  acting  upon  it  form  a  system  in  equilibrium. 


124  UNIFORM  FLOW  IN   OPEN  CHANNELS. 

Let  these  forces  be  resolved  in  the  direction  of  the  flow.  The 
forces  are  the  following: 

(a)  The  weight  of  the  body,  equal  to  wFl,  F  being  the 
cross-sectional  area  and  /  the  length  AB. 

(6)  The  pressures  of  the  adjacent  water  upon  the  cross- 
sections  at  A  and  B.  Since  the  two  sections  are  alike  in  all 
respects,  and  since  the  upper  surface  is  under  uniform  pressure, 
the  total  pressure  upon  the  cross-section  A  is  equal  and  oppo- 
site to  that  upon  B,  each  having  the  value  pF,  the  product 
of  the  cross-sectional  area  into  the  intensity  of  pressure  at  the 
centroid  of  the  area. 

(c)  The  frictional  forces  exerted  in  the  direction  BA  upon 
the  water  flowing  over  the  channel  surface.  Computing  it  as 
if  all  particles  of  the  water  had  equal  velocities,  and  assuming 
the  friction  per  unit  area  Pf  to  be  independent  of  the  pressure 
and  proportional  to  the  square  of  the  velocity,  the  total  fric- 
tional force  would  be 


C  being  the  length  of  the  wetted  perimeter,  and  k  a  constant 
depending  upon  the  roughness  of  the  surface. 

(d)  The  frictional  retardation  due  to  air  at  the  upper  sur- 
face. This  is  assumed  to  be  negligible. 

Equating  to  zero  the  sum  of  the  resolved  parts  of  these 
forces  in  the  direction  AB,  and  representing  by  s  the  sine  of 
the  angle  between  the  water  surface  and  the  horizontal,  we 
have 

wFls-kClv2  =  Q. 

Introducing  r=  F/C  =  hydraulic  radius  of  the  cross-section,  the 
equation  may  be  written 


in  which  c  is  a  coefficient  whose  value  depends  upon  the  rough- 
ness of  the  channel,  and  would  be  constant  for  any  given  kind 
of  channel  if  the  assumptions  above  made  were  rigorously  true. 
In  comparing  this  result  with  that  previously  obtained  for 
pipes  (Art.  113),  it  will  be  noticed  that  s  here  means  the  slope 


VARIATION  OF  COEFFICIENT  IN  CHEZY    FORMULA.      125 

of  the  water  surface,  while  in  the  previous  case  it  means  the 
hydraulic  slope.  It  is  obvious,  however,  that  the  hydraulic 
slope  of  a  stream  having  a  free  upper  surface  is  identical  with 
the  slope  of  that  surface.  If  a  piezometer  be  introduced  at  any 
cross-section,  as  in  Fig.  67,  the 
water  will  rise  in  it  to  the  level  of  r- 
the  surface  of  the  stream.  Fig.  66  Tf 

shows  at  once  that 

* "  i 


-= 
i       i' 

so  that  the  above  equation  might  be  written  in  either  of  the 
two  forms  given  in  Art.  113. 

129.  Variation  of  Coefficient  in  Chezy  Formula. — The  above 
theory  is  obviously  seriously  defective,  but  no  general  formula 
having  a  better  basis  in  theory  has  been  proposed.  In  accepting 
this  formula  as  a  practical  guide,  the  coefficient  c  must  be  regarded 
as  dependent  not  merely  upon  the  character  of  the  channel 
surface,  but  upon  various  other  conditions.  The  practical  rules 
which  have  been  given  for  estimating  the  value  of  c,  based 
upon  experiment,  usually  express  it  as  a  function  of  one  or 
more  of  the  three  quantities  v,  r,  and  s,  in  addition  to  the  rough- 
ness of  the  channel.  It  is  altogether  improbable  that  any  such 
assumption  can  properly  be  made  (except  as  a  rough  approxi- 
mation) in  comparing  streams  having  very  different  forms  of 
cross-section,  since  the  form  of  the  section  probably  affects  the 
flow  in  a  manner  which  is  not  dependent  merely  upon  the  value 
of  r,  and  which  in  fact  cannot  be  expressed  accurately  in  any 
simple  way.  In  solving  practical  problems  in  flow,  however, 
a  rough  approximation  to  the  true  result  is  often  all  that  it  is 
possible  to  attain. 

In  spite  of  the  defects  in  the  theory  above  given,  it  is  still 
true  that  the  most  important  factor  affecting  the  value  of  c  is 
the  character  of  the  channel  as  regards  roughness.  The  range 
of  values  of  the  coefficient  for  different  cases  may  be  stated 
roughly  as  follows : 


126 


UNIFORM   FLOW  IN  OPEN  CHANNELS. 


c  =  100  to  140  for  timber  or  cement-lined  conduits. 

c=   80  "   110    "   conduits  of  smooth  masonry. 

c=   60  "     90   "    conduits  of  rough  masonry  (rubble). 

c=  50  "     80  "    ditches  in  clean  earth  or  gravel. 

c=  20  "     40   "    ditches  or  canals  in  bad  order. 

The  experimental  determination  of  c  is  much  more  difficult 
for  open  channels  than  for  pipes.  An  accurate  determination 
requires  that  the  channel  under  experiment  shall  be  quite  truly 
uniform  in  slope;  otherwise  the  cross-section  of  the  stream 
will  vary  even  if  that  of  the  channel  does  not.  Unless  the 
condition  of  uniform  flow  is  very  accurately  maintained,  no 
reliable  value  of  c  can  be  deduced  from  the  experiment. 

130.  Experiments  of  Bazin.  —  An  extensive  series  of  experi- 
ments was  performed  by  Bazin  for  the  purpose  of  determining 
how  the  flow  is  influenced  by  the  character  of  the  surface,  the 
slope  of  the  channel,  and  the  size  and  shape  of  the  cross-section. 
From  these  experiments  the  author  drew  the  conclusion  that 
the  value  of  c  varies  but  little  with  s,  and  that,  so  far  as  the 
dimensions  of  the  cross-section  affect  it,  c  is  mainly  a  function 
of  r.  The  conclusions  reached  were  embodied  in  a  formula 
equivalent  to  the  following: 


The  empirical  constants  cf  and  a  depend  upon  the  roughness 
of  the  channel,  and  Bazin  gave  for  four  typical  cases  the  values 
shown  in  the  accompanying  table. 

TABLE  V. 

VALUES  OF  COEFFICIENTS  IN  FORMULA  OF  BAZIN  FOR  OPEN  CHANNELS. 


Kind  of  Channel. 

c' 

a 

Very  smooth  .  .     .    .            '  •  • 

143 

0.10 

(Cement  or  planed  timber.) 
Smooth                                                              

127 

0.23 

(Ashlar,  brickwork,  rough  timber.) 
Rough                                              

113 

0.82 

(Rubble  masonry.) 
Very  rough                                    

104 

4.10 

(Ditches  or  canals  in  bad  order.) 

KUTTER'S  FORMULA.  127 

It  will  be  noticed  that  c'  is  the  limiting  value  approached  by 
c  as  r  increases.  The  range  of  the  experiments  does  not,  how- 
ever, justify  the  use  of  the  formula  for  values  of  r  greater  than 
2  or  3  feet.  , 

131.  Kutter's  Formula.  —  The  only  wholly  general  rule  for 
choosing  the  value  of  c  which  has  been  widely  used  is  the 
formula  proposed  by  Ganguillet  and  Kutter.  This  formula  is 
the  result  of  a  careful  study  of  all  the  experimental  data  regard- 
ing flow  in  open.  channels  which  was  known  to  its  authors.  It 
is  designed  to  apply  to  the  widest  range  of  cases,  from  the 
smallest  of  artificial  channels  up  to  large  rivers,  and  to  take 
account  of  all  factors  affecting  the  value  of  the  coefficient. 

In  Kutter's  formula  c  is  expressed  in  terms  of  the  slope  s, 
the  hydraulic  radius  r,  and  a  third  quantity  n  called  the  coeffi- 
cient of  roughness.  For  English  units  the  expression  is 


n 


The  authors  of  the  formula  gave  values  of  n  for  six  typical 
kinds  of  channel  surface,  ranging  from  .010  for  flumes  lined 
with  well-planed  timber  or  neat  cement  to  .030  for  streams 
impeded  by  detritus  or  aquatic  plants.  The  following  more 
detailed  series  of  values  is  often  given : 

Nature  of  Channel. 

n 

Well-planed  timber 009 

Neat  cement 010 

Cement  one-third  sand. Oil 

Ashlar  and  brickwork 013 

Canvas  on  frames 015 

Rubble  masonry 017 

Canals  in  very  firm  gravel 020 

Rivers  and  canals  in  perfect  order,  free  from  stones  or  weeds .025 

Rivers  and  canals  in  moderately  good  order 030 

Rivers  and  canals  in  bad  order,  with  weeds  and  detritus 035 

Torrential  streams  encumbered  with  detritus ' 050 

While  these  values  may  be  fairly  reliable  for  channels  of 
uniform  slope  and  alignment,  greater  values  should  probably 


128  UNIFORM  FLOW  IN  OPEN   CHANNELS. 

be  used  in  designing  flumes  and  ditches  under  ordinary  prac- 
tical conditions.  The  range  of  values  given  in  Fig.  69  may 
be  suggested. 

132.  Reliability   of   Kutter's    Formula.  —  This    formula   has 
gained  wide  acceptance  as  a  guide  in  choosing  values  of  c  for 
all  cases  of  open  channels,  and  also  for  large  pipes  conveying 
water  under  pressure.     It  is  undoubtedly  a  formula  of  great 
value,  since  it  provides  a  rule  applicable  to  cases  for  which 
otherwise  there  would  be  no  guide  whatever  because  of  the 
lack  of  experimental  data.     The  student  should,  however,  be 
warned  against  the  too  implicit  acceptance  of  the  results  of 
this  or  any  other  empirical  formula  in  hydraulics,  and  espe- 
cially against  the  supposition  that  the  estimated  values  of  c 
(and  the  quantities  computed  from  it)  can  be  regarded  as  more 
than  approximations  subject  to  a  considerable  percentage  of 
uncertainty.    It  is  quite  common  for  writers  to  give  values  of 
c  to  four  significant  figures,  when  in  fact  they  can  hardly  be 
supposed  to  be  certainly  correct  to  within  10  per  cent.    Here 
as  elsewhere  in  engineering  computations  it  is  important  for 
the  student  to  acquire  the  habit  of  putting  a  reasonable  esti- 
mate on  the  degree  of  reliability  of  his  data  and  formulas. 

133.  Tables  Computed  from  Kutter's  Formula.  —  To  facilitate 
the  use  of  Kutter's  formula,  values  of  c  computed  from  it  are 
given  in  Table  VI.    Values  not  given  directly  in  the  table  may 
be  found  with  sufficient  accuracy  by  interpolation. 

134.  Graphical    Representation   of  Kutter's   Formula.  —  The 
formula  of  Ganguillet  and  Kutter  may  be  written  in  the  form 

y       _  yVr 


in  which 


TABLE  COMPUTED  FROM  KUTTER'S  FORMULA. 


129 


TABLE    VI. 

VALUES  OF  c  IN  THE  FORMULA  v  =  c\/rs,  COMPUTED  FROM  FORMULA  OF 
GANGUILLET  AND  KUTTER. 


8 

n 

r  (feet). 

.2 

.3 

.4 

.6 

.8 

1.0 

1.5 

2.0 

3 

4 

6 

8 

10 

15 

.00005 

.010 

87 

99 

109 

122 

133 

140 

154 

164 

178 

187 

199 

206 

212 

222 

.012 

68 

79 

88 

98 

107 

114 

126 

135 

148 

156 

168 

175 

181 

190 

.015 

51 

59 

66 

76 

83 

89 

99 

107 

118 

126 

137 

144 

149 

158 

.020 

35 

41 

46 

53 

59 

64 

72 

79 

88 

95 

105 

111 

116 

125 

.025 

26 

31 

35 

41 

46 

49 

57 

62 

71 

77 

85 

91 

96 

104 

.030 

21 

25 

28 

33 

37 

40 

47 

51 

59 

64 

72 

78 

82 

90 

.035 

18 

21 

24 

28 

31 

34 

40 

44 

50 

56 

63 

68 

72 

80 

.040 

15 

18 

20 

24 

27 

29 

34 

38 

44 

49 

56 

61 

64 

72 

.0001 

.010 

98 

109 

119 

131 

140 

147 

159 

168 

178 

186 

195 

201 

205 

212 

.012 

76 

87 

95 

105 

114 

120 

130 

138 

149 

155 

164 

170 

174 

181 

.015 

57 

65 

72 

81 

88 

93 

103 

109 

119 

125 

134 

139 

143 

150 

.020 

39 

45 

50 

57 

63 

67 

75 

81 

89 

94 

102 

107 

111 

118 

.025 

29 

34 

38 

44 

48 

52 

59 

64 

71 

76 

84 

88 

92 

98 

.030 

23 

27 

31 

35 

39 

42 

48 

53 

59 

64 

71 

75 

78 

85 

.035 

19 

22 

25 

30 

33 

35 

41 

45 

51 

55 

61 

66 

69 

75 

.040 

16 

19 

22 

25 

28 

31 

35 

39 

45 

49 

54 

59 

62 

68 

.0002 

.010 

105 

116 

125 

138 

145 

151 

162 

170 

179 

185 

193 

198 

201 

207 

.012 

83 

92 

100 

111 

118 

123 

133 

140 

149 

155 

162 

167 

170 

176 

.015 

61 

69 

76 

85 

91 

96 

105 

111 

119 

125 

132 

137 

140 

145 

.020 

42 

48 

53 

60 

65 

69 

77 

82 

89 

94 

100 

105 

108 

113 

.025 

31 

36 

40 

46 

50 

54 

60 

64 

72 

76 

82 

87 

89 

95 

.030 

25 

29 

32 

37 

41 

44 

49 

54 

59 

63 

69 

73 

76 

82 

.035 

21 

24 

27 

31 

34 

37 

42 

45 

51 

55 

60 

64 

67 

72 

.040 

17 

20 

23 

26 

29 

32 

36 

40 

45 

48 

53 

57 

60 

65 

.0004 

.010 

110 

120 

129 

140 

148 

154 

164 

170 

179 

184 

191 

196 

199 

204 

.012 

87 

96 

104 

113 

121 

125 

135 

141 

149 

154 

161 

165 

168 

174 

.015 

65 

73 

79 

87 

93 

98 

106 

112 

119 

124 

130 

135 

138 

143 

.020 

44 

50 

55 

62 

67 

70 

78 

83 

89 

94 

99 

104 

107 

112 

.025 

32 

37 

42 

47 

51 

55 

60 

65 

71 

76 

81 

85 

88 

94 

.030 

25 

30 

33 

38 

42 

45 

50 

54 

59 

63 

69 

73 

75 

81 

.035 

21 

24 

27 

31 

35 

37 

42 

45 

51 

55 

60 

64 

66 

72 

.040 

18 

21 

23 

27 

30 

32 

37 

40 

45 

48 

53 

57 

59 

65 

.0010 

.010 

113 

124 

131 

142 

150 

155 

165 

171 

179 

184 

190 

194 

197 

202 

.012 

89 

98 

105 

115 

122 

127 

136 

142 

149 

154 

160 

164 

167 

172 

.015 

66 

74 

80 

88 

94 

99 

108 

112 

119 

124 

130 

134 

136 

141 

.020 

45 

51 

56 

63 

68 

71 

78 

83 

89 

93 

99 

103 

105 

110 

.025 

34 

39 

43 

48 

52 

56 

62 

66 

71 

75 

81 

85 

87 

92 

.030 

27 

30 

34 

39 

42 

45 

50 

54 

59 

63 

68 

72 

74 

79 

.035 

22 

25 

28 

32 

35 

3« 

43 

46 

51 

54 

59 

63 

65 

70 

.040 

18 

21 

24 

27 

30 

33 

37 

40 

45 

48 

52 

56 

58 

63 

130 


UNIFORM   FLOW  IN   OPEN   CHANNELS. 


If  s  be  eliminated  between  these  equations,  there  results  an 
equation  between  x,  y,  and  n.  For  a  given  constant  value  of 
n  this  equation  may  be  represented  by  a  curve  with  x  and  y 
as  rectangular  coordinates.  A  series  of  such  curves  may  thus 
be  drawn,  each  corresponding  to  a  definite  value  of  n.  In  like 
manner  a  second  series  of  curves  may  be  determined,  each 
corresponding  to  a  definite  value  of  s. 
Thus,  eliminating  s, 


x=ny -1.811. 


Eliminating  n, 


.00281 


(4) 


(5) 


Taking  rectangular  axes  OX,  OY  (Fig.  68),  let  equation  (4) 
be  plotted  for  some  definite  value  of  n.    The  resulting  locus  is 

1  811 
a  straight  line  AB,  such  that  OA  =  1.811,  OB  =  - .     For  a 

Tl 

second  value  of  n  another  straight  line  is  obtained  passing 
through  the  same  point  A. 

Next  consider  the  curve  represented  by  (5)  when  s  has  any 
constant  value.    This  is  seen  to  be  a  rectangular  hyperbola 

Y 


Q    0 
FIG.  68. 


M 


whose  asymptotes  are  OY  and  a  line  parallel  to  OX  at  a  dis- 

00981 
tance  from  it  0^=41.65  +  ^=^. 

s 

Let  lines  such  as  AB  be  drawn  for  a  series  of  values  of  nf 
and  curves  such  as  CD  for  a  series  of  values  of  s.    The  point 


CHANNEL  OF  SMALLEST  CROSS-SECTION.  131 

in  the  plane  OXY  which  corresponds  to  any  definite  values  of 
n  and  s  can  then*  be  located  by  inspection.  The  values  of  x 
and  y  are  thus  known,  and  c  may  be  found  by  the  following 
graphical  construction. 

For  any  given  value  of  r,  let  OM  (Fig.  68)  be  taken  equal 
to  vV.  Let  N  be  the  point  whose  co-ordinates  are  x,  y,  deter- 
mined from  any  given  values  of  n  and  s.  Draw  MN,  and  let 
P  be  its  point  of  intersection  with  OY.  From  similar  triangles, 

OP    OM 


Qp_OMxQN_jV7 

QM        Vr+x 

Comparing  with  (1), 


From  such  a  diagram  it  is  possible  not  only  to  determine  c 
when  r,  s,  and  n  are  given,  but  to  determine  any  one  of  the 
four  quantities  when  the  other  three  are  given.  For  accurate 
computations  the  diagram  should  be  drawn  carefully  to  a  large 
scale.  But  even  the  diagram  shown  in  Fig.  69  suffices  for  mak- 
ing computations  with  less  uncertainty  than  that  of  the  data 
usually  involved  in  practical  hydraulic  problems.* 

135.  Smallest   Cross-section   for   Given   Rate   of  Discharge. 

—  If,  with  a  given  slope,  a  channel  is  to  discharge  water  at  a 
given  rate,  the  cross-sectional  area  will  depend  upon  the  form 
of  section  adopted.    Let  it  be  required  to  make  this  area  as 
small  as  possible. 
In  the  equation 

_  _  ET| 

q  =  Fv  =  Fc\/rs  =  c\/«  — 

the  conditions  of  the  problem  make  s  constant,  and  it  will  be  as- 
sumed that  c  is  constant.  Then  q  being  expressed  as  a  function 
of  two  variables  F  and  C,  it  is  required  to  satisfy  the  conditions 

q  =  constant,        F  =  a  minimum. 

*  The  computation  of  velocity  for  any  values  of  r  s,  and  n  is  greatly  facil- 
itated by  the  use  of  diagrams  published  by  Prof.  I.  P.  Church. 


132 


UNIFORM  FLOW  IN   OPEN  CHANNELS. 


c. 

T-H 

T-H 

s 

<N 

0       0 

"8 

0 

0 

o 

0 

o     o 

1 

l" 

0 

r 

1 

CO 

r 
t> 

r    r 

"3 

T-H 

0 

o 

o 

T-H 

0 

§    o 

J 

1 

]3 

Q 

§ 

| 

g 

bfi 

8 

8 

•a 

I 

a 

j 

o. 

1 

2 

-£ 

o 

i 

h 

O 

0 

VH 

i 

b 

s    ® 

1 

i 

1 

.0 

S 

s  mason 

J  .g 

o 

.3    | 

•i 

1 

3 

3 

i  -i 

g 

§ 

a 

? 

JD 

a 

S      e 

5  a 

I      i      I  I I 


jo        »         S9n[t?^       w 
I      i      I      i      i      I      l      I 


I      l      I      I 


CHANNEL  OF  SMALLEST  CROSS-SECTION.  133 

From  the  form  of  the  above  value  of  q  it  follows  that,  q  being 
constant,  C  is  a  minimum  when  F  is. 

The  values  of  F  and  C  admit  of  infinite  variation,  depend- 
ing upon  the  form  chosen  for 
the  cross-section.  The  discussion 
will  be  restricted  to  the  case  of  a 
trapezoidal  section,  represented  in 
Fig.  70. 

Trapezoidal     section.  —  Let    x=  FIG.  70. 

bottom   width,  y  =  depth   of  water, 

6  =  angle  between  side  slope  and  horizontal;  then  F  and  C  can 
be  expressed  in  terms  of  x  and  y.  For  any  fixed  value  of  0 
let  it  be  required  to  determine  what  relation  between  x  and  y 
will  give  minimum  values  of  F  and  p. 

We  have  F  =  y(x  +  y  cotan  6)  ; 

C  =  x  +  2y  cosec  6. 

The  conditions  to  be  satisfied  are 


Differentiating, 

dF  =  ydx  +  (x  +  2y  cotan 
dC  =  dx  +  2  cosec  6-dy  =   . 

Eliminating  dy/dx  between  these  equations, 

2(1-0080) 

ff  =  _  _  _  L  f>/ 

sin  0      ^ 

which  is  the  relation  between  x  and  y  for  minimum  cross-sec- 
tion and  wetted  perimeter.  By  means  of  this  relation  the 
values  of  F,  C,  and  r  may  be  expressed  in  the  forms 


cotan  6)  =  (2  cosec  6  —  cotan  d)y2, 
C=x  +  2y  cosec  6  =  2(2  cosec  6  -  cotan  6)  y, 


134  UNIFORM  FLOW  IN   OPEN   CHANNELS. 

Rectangular  section.  —  If  0  =  90°,  the  trapezoid  reduces  to  a 
rectangle,  and  the  solution  for  minimum  cross-section  and 
wetted  perimeter  becomes  « 


EXAMPLES. 

In  the  following  examples  let  c  be  determined  by  Kutter's  formula. 

1.  A  rectangular  flume  of  rough  plank  is  to  be  laid  on  a  slope  of 
1/1000,  and  is  to  discharge  10  cu.  ft.  per  sec.     Estimate  the  width  and 
depth  for  minimum  cross-section  and  wetted  perimeter. 

Ans.  y  =  1.3',x=2.G'. 

2.  A  ditch  with   trapezoidal  section,  side  slopes  45°,  slope  of  bed 
1/1000,  is  to  discharge  40  cu.  ft.  per  sec.     Required  the  best  values  of 
the  width  and  depth.     Assume  n  =  .025.  Ans.  y=3.6'. 

3.  Compute  the  rate  of  discharge  of  a  rectangular  flume  of  brick 
masonry,  3.5  ft.  wide,  sloping  1  ft.  in  3000  ft.,  the  depth  of  the  water 
being  2  ft.  Ans.  About  15.8  cu.  ft.  per  sec. 

4.  Prove  that,  of  all  trapezoidal  sections  giving  a  certain  rate  of  dis- 
charge, the  section  of  minimum  area  has  side  slopes  of  60°. 


CHAPTER  XII. 
OPEN  CHANNELS:  NON-UNIFORM  FLOW. 

136.  General    Equation  of   Energy  for  Stream    of  Variable 
Cross-section. — The  reasoning  by  which  the  general  equation 


~ 
w     2g 


- 
w     2g 


was  established  (Arts.  59,  60)  is  valid  in  the  case  of  a  stream 
with  a  free  upper  surface.  For  such  a  stream  the  hydraulic 
gradient  coincides  with  the  surface  of  the  stream,  and  z  -f  p/w  or 
y  denotes  the  height  of  the  water  surface  above  a  horizontal 
datum  plane.  (See  Fig.  67.) 

In  Fig.  71  let  A  and  B  be  any  two  cross-sections  of  a  stream, 
the  flow  being  from  A  toward       A 
B]    let  2/1,  vi  be  the  values  of 
y  and  v  at  A}  and  y2,  v2  the  values 
at  B]    and  let  Hf  denote  the 
loss  of  head  between  A  and  B 
(i.e.,  the  energy  lost  between  A 
and  B  for  every  pound  of  water 
passing  any  cross-section  of  the  FIG.  71. 

stream) .     Then 

/  11, 2\         /  ,,^2\ 

-H' (1) 

137.  Value  of  Lost  Head. — In  the  case  of  uniform  flow  the 
loss  of  head  in  any  given  length  I  of  the  stream  may  be  ex- 
pressed by  means  of  the  formula 

v  =  cVrs. 

135 


136  OPEN  CHANNELS:   NON-UNIFORM  FLOW. 

For  in  this  case,  Vi  and  v2  being  equal, 


and 


so  that 


c2r 


(2) 


Let  this  same  expression  for  Hf  be  assumed  for  the  case  of 
variable  flow,  it  being  understood  that  v  and  r  have  values 
intermediate  between  those  applying  to  the  sections  A  and  B. 
Equation  (1)  then  becomes 


138.  Differential  Equation  for  Surface  Curve.  —  Consider  a 

longitudinal  section  of  the  stream  by  a  vertical  plane,  and  let 

x,  y  be  the  coordinates  of  the  surface  curve  referred  to  axes 

Xj  OY,  the  former  being  horizontal  and  the  latter  vertical 

Y 


FIG.  72. 


(Fig.  72).  If  the  above  equation  be  applied  to  the  portion  of 
the  stream  between  two  sections  A  and  B  infinitely  near 
together,  we  must  put 

l=dx,    y2-yi=dy, 


CHANNEL   OF  UNIFORM  SHAPE  AND  SLOPE,  137 

so  that  the  equation  becomes 

v          v2 


-0  (4} 

\Jb  V         I  *->       \AS**S  V/»  •  •  •  •  •  «  \^7 

g         czr 

139.  Channel  of  Uniform  Shape  and  Slope. — Let  the  channel 
have  a  uniform  slope  and  uniform  cross-section,  the  cross-sec- 
tion of  the  stream,  however,  varying  with  the  depth  of  the 
water.  Let 

i  =  slope  of  bed ; 

u  =  depth  of  stream  at  point  (x,  y) ; 

b  =  width  of  stream  at  water  surface. 

Let  y  be  replaced  by  u,  by  means  of  the  evident  relation 

dy  =  du-i  dx. 
Also,  we  have 

v  =  -^  (q  being  constant) ; 


q  dF  _       qbdu 
~      =       ~ 


Equation  (4)  may  therefore  be  written 


Now  let  q  be  replaced  by  its  value  in  terms  of  the  dimen- 
sions of  a  uniform  stream  in  the  same  channel.  Thus,  if  the 
channel  were  unobstructed  for  a  great  distance,  the  surface 
would  assume  some  position  as  MN  (Fig.  72)  parallel  to  the 
bed.  Let  u',  F',  r'  be  the  values  which  u,  F,  r  would  have  in 
such  a  case;  then 


and  equation  (5)  takes  the  form 

(^      r'F/2V,7       /i      c*i, 

(I-  —)tdx={i-.—-du  .....  (6) 


138  OPEN  CHANNELS:    NON-UNIFORM   FLOW. 

When  the  form  of  the  channel  is  given,  F,  b  and  r  can  be  ex- 
pressed in  terms  of  u,  and  the  relation  between  u  and  x  is  then 
to  be  found  by  integrating  equation  (6) .  In  certain  ideal  cases 
the  integration  is  fairly  simple.  The  one  most  commonly 
treated  is  that  in  which  the  cross-section  is  assumed  to  be  of 
uniform  depth  and  of  great  width. 

140.  Cross-section   of   Great  Width   and   Uniform   Depth. — 
For  a  rectangular  cross-section  b  is  constant  and 


u'  being  the  depth  corresponding  to  cross-section  F'. 

If  b  is  very  great  in  comparison  with  u,  we  have  approxi- 
mately 

r  =  u,     r'  =  u', 

and  equation  (6)  becomes 


Let  u/u'=z;  then  du  =  u'dz,  and 


-  ,                     „  i\    dz 
or  -,dx  =  dz+(l-— )  — - (8) 

For  brevity  let 


23-l 

then  the  integration  of  (8)  gives 


9  jT  /  (*   "9  \ 

—  =  z-  (1  -  — }  (f)(z)  +  constant (9) 


BACKWATER. 


139 


Let  the  integration  be  taken  between  limits  x*=x\  and 
x  =  X2,  where  x%  —  o;i  =  /  =  distance  between  any  two  definite 
cross-sections  of  the  stream,  zlt  z2  being  the  corresponding 
values  of  z.  Then 


.    .    .    .  (10) 


In  order  to  apply  this  equation  it  is  necessary  to  compute 
values  of  <j>(z)  for  a  series  of  values  of  z.  The  general  value  of 
the  function  is  found  by  integration  *  to  be 


r   dz       1 

>  -J  ^r6 


From  this  are  computed  the  values  entered  in  Table  VII. 

For  very  exact  computations  a  fuller  table  is  required. 
Intermediate  values  may,  however,  be  found  by  interpolation 
with  less  error  than  that  due  to  the  defects  in  the  above  theory 
and  the  uncertainty  always  existing  in  the  data  of  hydraulic 
problems. 

TABLE  Vll. 

=  -  /  -;  —  7. 
J  z  —  i 


VALUES  OF 


z 

«.) 

z 

*.) 

z 

to) 

z 

*(.) 

.00 

oo 

1.10 

.680 

1.30 

.373 

1.65 

.203 

.01 

1.419 

1.12 

.626 

1.32 

.357 

1.70 

.189 

.02 

1.191 

1.14 

.581 

1.34 

.342 

1.80 

.166 

.03 

1.060 

1.16 

.542 

1.36 

.328 

1.90 

.147 

.04 

.967 

1.18 

.509 

1.38 

.316 

2.00 

.1318 

.05 

.896 

1.20 

.480 

1.40 

.304 

2.10 

.1188 

.06 

.838 

1.22 

.454 

1.45 

.278 

2.20 

.1074 

.07 

.790 

1.24 

.431 

1.50 

.255 

2.30 

.0978 

1  08 

.749 

1.26 

.410 

1.55 

.235 

2.40 

.0894 

1  09 

713 

1  28 

.390 

1.60 

.218 

2  50 

.0822 

141.  Backwater.— If  the  surface  of  a  stream  at  a  certain 
point  be  raised  by  a  dam  or  other  obstruction,  the  effect  will 
extend  up-stream  for  some  distance,  decreasing  as  the  distance 

*  See  Williamson's  Integral  Calculus,  p.  59.  To  the  value  there  given 
is  added  such  a  constant  that  <p(z)  =  Q  when  2=00. 


140  OPEN  CHANNELS:    NON-UNIFORM  FLOW. 

from  the  obstruction  increases.  To  estimate  the  amount  by 
which  the  surface  is  raised,  at  a  given  distance  above  the  darn, 
is  a  practical  question  often  submitted  to  the  hydraulic  engi- 
neer. If  the  stream  is  broad  and  shallow,  and  if  the  bed  has  a 
fairly  uniform  slope,  the  foregoing  theory  may  be  applied.  The 
method  of  procedure  may  be  illustrated  by  the  following  ex- 
ample. 

Suppose  a  stream  is  originally  20  ft.  deep,  and  that  the 
surface  is  raised  by  an  obstruction  so  that  the  depth  at  a  cer- 
tain point  becomes  30  ft.  Let  the  slope  of  the  bed  be  1/5000 
and  let  c  =  80.  It  is  required  to  determine  at  what  distance 
from  the  given  section  the  increase  in  depth  is  2  ft. 

From  the  given  data  u' =  20,  t  =  .0002,  c  =  80,  22  =  1.5,  21  = 
1.1.  The  table  gives  </>(zO  =  .680,  </>(z2)  =  .255,  hence 

Z  =  100000[(1.5-l.l)  +  960(.680-. 255)]  =  80800  ft. 

If,  instead  of  z\,  I  is  given  and  z\  is  required,  the  solution 
is  not  so  direct.  Thus,  suppose  in  the  above  case  it  is  required 
to  determine  the  increase  in  depth  at  a  section  25000  ft.  above 
the  point  where  the  depth  is  30  ft.;  that  is,  Z= 25000  and  z\ 
is  required.  Substituting  in  equation  (10), 

«i-.960^(2i)  =  1.005. 

By  taking  trial  values  from  the  table  it  is  found  that  the  equa- 
tion is  nearly  satisfied  by  z\  =  1.34.  The  depth  at  the  specified 
section  is  therefore  1.34x20  ft.  =26.8  ft. 

In  applying  the  above  theory  to  actual  streams  for  which 
the  assumption  of  uniform  depth  throughout  the  cross-section 
is  not  even  roughly  true,  u  may  be  taken  as  the  average  depth 
for  the  cross-section,  computed  from  the  formula  u  = 
This  method  may  be  applied  to  Ex.  4  of  the  following  list. 


BACKWATER.  141 

EXAMPLES. 

1.  A  wide  stream  12  ft.  deep  is  obstructed  so  that  the  depth  at  a 
certain  section  becomes  16  ft.     The  slope  of  the  bed  is  2  ft.  per  mile. 
Compute  the  depth  (a)  2  miles  up-stream  and  (b)  2  miles  down-stream. 
[Takec=65.J  Ans.  (a)  14.0ft.     (6)  18.7ft. 

2.  In  Ex.  1,  determine  the  positions  of  sections  at  which  the  depth 
has  values  13  ft.,  15  ft.,  and  16  ft. 

Ans.  The  depth  would  be  13  ft.  at  about  3.73  miles  up-stream. 

3.  A  wide  stream  20  ft.  deep  is  obstructed  so  that  the  depth  at  a 
certain  section  becomes  25  ft.     If  i  =  .0002  and  c  =  70,  determine  (a)  the 
depth  10000  ft.  up-stream  from  the  given  section,  and  (6)  where  the 
depth  will  be  21  ft.     Ans.  (a)  24.1ft.     (6)  About  66000ft.  up-stream. 

4.  A  stream  whose  cross-sectional  area  is  5400  sq.  ft.  and  width  350 
ft.  when  unobstructed  has  its  surface  raised  10  ft.  at  a  certain  section 
by  a  dam.     The  slope  is  1.6  ft.  per  mile,  and  the  value  of  c  is  75.     (a) 
Compute  the  rise  of  the  surface  at  distances  of  2  miles  and  5  miles  above 
the  given  section,     (b)  Where  will  the  increase  of  depth  be  5  ft.? 

Ans.  (a)  7.6  ft.  and  4.6  ft.     (b)  At  4.53  miles  up-stream. 


CHAPTER  XIII. 
THE  MEASUREMENT  OF  RATE  OF  DISCHARGE. 

142.  General  Methods. — One  of  the  most  important  of  the 
problems  of  practical  Hydraulics  is  the  determination  of  the 
rate  of  discharge  of  streams.     The  conditions  of  the  problem 
vary  greatly  in  different  cases,  and  the  methods  employed  must 
vary  correspondingly.     The  quantities  to  be  measured  vary  from 
the  discharge  of  a  small  pipe  to  that  of  a  large  river. 

The  methods  employed  may  be  classed  as  (1)  direct  and 
(2)  indirect. 

(1)  An  actual  measurement  may  be  made  of  either  (a)  the 
total  quantity  discharged  in  a  given  time,  or  (6)  the  velocity  of 
'flow. 

(2)  Measurement  may  be  made  of  certain  quantities  upon 
which  the  rate  of  discharge  is  known  to  depend,  its  value  being 
computed  from  the  data  thus  determined. 

143.  Direct    Measurement    of   Total  Discharge. — The    total 
quantity  discharged  in  a  given  time  may  be  determined  either 
by  weighing  the  water  or  by  measuring  its  volume  in  a  prop- 
erly calibrated  vessel  or  reservoir. 

Accurate  measurement  by  weighing  will  in  general  be  pos- 
sible only  when  the  rate  of  discharge  is  small,  since  for  accurate 
results  the  experiment  must  extend  over  a  considerable  time. 
A  discharge  of  1  cu.  ft.  per  second  would  give  in  1  minute  a 
total  discharge  of  3750  Ibs. 

If  a  reservoir  is  available  whose  volume  for  given  depths  is 
accurately  known,  the  total  discharge  in  a  given  time  may  be 
measured  by  collecting  the  water  in  this  reservoir.  Much  larger 
quantities  can  thus  be  measured  than  it  is  practicable  to  weigh. 

142 


METHODS  OF  MEASURING  VELOCITY.  143 

This  method  is  sometimes  employed  in  estimating  the  rate  of 
discharge  of  the  conduit  of  a  water  supply  system,  the  discharge 
for  a  period  of  several  hours  or  even  a  day  or  more  being  col- 
lected in  a  large  storage  reservoir  whose  horizontal  area  at  dif- 
ferent levels  is  accurately  known.  Such  measurements  are 
liable  to  uncertainty  because  of  evaporation  and  leakage,  and 
also  because  a  small  error  in  measuring  the  depth  of  the  water 
results  in  a  relatively  large  error  in  the  total  quantity. 

144.  Discharge  Computed  from  Direct  Measurement  of 
Velocity.  —  If  the  velocity  of  flow  be  measured  at  many  differ- 
ent points  in  a  given  cross-section  of  a  stream,  the  rate  of  dis- 
charge for  the  whole  section  may  be  closely  estimated.  Let 
the  total  cross-section  be  F}  and  let  it  be  divided  into  parts 
whose  areas  are  FI,  F2,  .  .  .  ,  the  values  of  the  mean  velocity 
in  these  parts  being  i?1;  v2,  .  .  .  ,  and  the  mean  velocity  for  the 
entire  section  v.  Then  the  total  rate  of  discharge  is 


This  method  is  especially  applicable  to  open  streams  of  consid- 
erable size. 

145.  Methods  of  Measuring  Velocity.  —  Of  the  various  devices 
that  have  been  employed  for  measuring  velocity  of  flow  there 
may  be  mentioned  floats,  the.  tachometer  or  current-meter,  and 
the  Pitot  tube. 

Floats  are  especially  applicable  to  streams  of  considerable 
size.  Under  favorable  conditions  they  may  be  used  for  measur- 
ing the  velocity  not  only  at  the  surface  but  at  any  desired 
depth,  being  suspended  by  wires  from  surface  floats  of  such 
shape  and  size  as  to  be  little  influenced  by  the  surface  velocity. 
Floats  are  also  sometimes  made  in  the  form  of  tubes  so  weighted 
as  to  float  in  an  upright  position  and  extend  from  the  surface 
nearly  to  the  bottom,  the  motion  of  a  float  thus  indicating  a 
mean  of  the  velocities  at  different  depths.  The  field  operations 
required  in  this  and  other  methods  of  gauging  the  flow  of  large 
streams  are  quite  elaborate  and  will  not  be  here  described.* 

*  Con  suit  Physics  and  Hydraulics  of  the  Mississippi  River,  by  Humphreys 
and  Abbot. 


144         THE  MEASUREMENT  OF  RATE  OF  DISCHARGE. 

The  current-meter  consists  of  a  vaned  wheel  which  is  rotat- 
ed by  the  current  at  a  rate  depending  upon  the  velocity.  The 
total  number  of  revolutions  in  a  given  time  is  counted,  usually 
by  means  of  an  automatic  register.  The  relation  between  the 
velocity  of  flow  and  the  number  of  revolutions  per  unit  time 
must  be  determined  for  any  given  instrument  by  experiment. 
This  is  called  "rating"  the  meter. 

Pitot's  tube,  in  its  simplest  form,  is  an  open  tube  one  end 
of  which  is  bent  at  right  angles  to  the  length  of  the  main  por- 
tion.    The  bent  end  is  submerged  and 
placed  with  the  opening  facing  the  cur- 
-  rent,  while  the  other  end  projects  above 


\  the  free  surface  (Fig.  73).     The  water 
=  in  the  tube  rises  above  the  surface  of 


-y_-f:£j  £|£^~:      :  the  stream  to  a  height  depending  upon 

^  the  velocity  of  flow  at  the  lower  end. 

¥IG.  73.  No  exact  relation  between  the  velocity 

v  and  the  height  h  of  the  column  above 

the  water  surface  can  be  determined  from  theory  alone.  It 
would  seem  that  the  increase  of  pressure  at  the  lower  end  of 
the  tube  will  be  proportional  approximately  to  the  square  of 
the  velocity,  so  that 


k  being  a  coefficient  whose  value  for  any  given  instrument  must 
be  determined  experimentally.     (See  Art.  171.) 

The  Pitot  tube  has  also  been  employed  for  measuring  veloci- 
ties in  pipes  carrying  water  under  pressure.  For  such  use  the 
instrument  consists  of  two  tubes  whose  lower  ends  are  open, 
but  so  placed  that  one  opening  faces  the  current,  while  the 
other  is  at  right  angles  to  it.  If  the  upper  ends  are  open,  water 
will  rise  in  the  two  tubes  to  different  heights,  the  difference 
being  a  measure  of  the  velocity  of  the  current.  If  the  pressure 
is  too  great  for  the  use  of  open  piezometers,  a  difference-gauge, 
such  as  is  shown  in  Fig.  36,  may  be  connected  with  the  two 
tubes  of  the  instrument.  Instead  of  mercury  a  fluid  of  less 
density  may  be  used  if  increased  sensitiveness  is  required. 


MEASUREMENT   BY   ORIFICES.  .  145 

146.  Indirect  Methods  of  Measuring  Rate  of  Discharge.  — 

Of  the  methods  which  above  were  called  indirect,  the  most 
important  are  three  :  by  orifices,  by  weirs,  and  by  the  Venturi 
meter.  These  will  be  considered  in  order. 

147.  Measurement  by  Orifices.  —  If  a  stream  be  conducted 
into  a  tank  or  reservoir  in  the  side  of  which  is  an  orifice,  the 
surface  of  the  water  will  assume  such  a  position  that  the  rate 
of  discharge  through  the  orifice  becomes  equal  to  the  rate  at 
which  water  flows  into  the  reservoir. 

The  rate  of  discharge  of  an  orifice  of  cross-section  F,  under 
a  head  h  on  its  center,  is 

q  =  cFV2gh,    .......     (1) 

c  being  the  coefficient  of  discharge  .  (Art.  44).  This  formula 
may  be  used  unless  the  dimensions  of  the  orifice  are  relatively 
large  in  comparison  with  the  head.  For  a  large  orifice  whose 
plane  is  not  horizontal  the  form  of  the  formula  depends  upon 
the  shape  and  cross-section  of  the  orifice,  as  shown  in  Art.  46. 
It  is  rarely  needful,  however,  to  employ  the  theoretically  more 
accurate  formula,  since  the  difference  between  the  results  given 
by  it  and  by  the  approximate  formula  is  usually  much  less  than 
the  uncertainty  in  the  value  of  the  coefficient  of  discharge. 
Thus  for  a  circular  orifice  in  a  vertical  plane  the  accurate 
formula  may  be  developed  in  a  series  of  which  the  first  two 
terms  are  as  follows: 


d  being  the  diameter.  If  d/h  =  l,  the  second  term  is  less  than 
one  per  cent  of  the  first,  and  the  series  converges  rapidly.  For 
a  rectangular  orifice  of  width  b  the  exact  formula  (Art.  49)  is 


W),  ......     (3) 

in  which  hi  and  h2  are  the  depths  of  the  upper  and  lower  edges 
below  the  water  surface.  Putting  h2  =  h+d/2  and  hi=h  —  d/2, 
this  can  be  expressed  as  a  series  of  which  two  terms  are  as 

follows  : 


146 


THE  MEASUREMENT  OF   RATE   OF  DISCHARGE. 


.     .     -     (4) 


If  d/h  =  l,  the  second  term  is  a  little  greater  than  one  per  cent 
of  the  first,  and  the  series  converges  quite  rapidly. 

The  coefficients  of  discharge  for  orifices  made  in  a  standard 
way  are  fairly  well  known  by  experiment.  The  accompanying 
tables  *  give  values  of  c  for  circular  standard  orifices,  square 
orifices,  and  rectangular  orifices  one  foot  wide.  In  every  case 
the  orifice  is  supposed  to  be  formed  with  a  sharp  inner  edge 
(as  at  A,  Fig.  17),  so  that  full  contraction  of  the  jet  occurs.  It 
is  also  needful  that  the  orifice  shall  be  at  a  considerable  distance 
from  the  sides  and  bottom  of  the  reservoir;  otherwise  the 
approaching  particles  of  water  will  be  so  guided  that  full  con- 
traction will  be  prevented  and  the  value  of  c  will  be  uncertain. 

The  horizontal  lines  in  the  tables  indicate  the  limiting  values 
of  h  below  which  the  approximate  formula  (1)  should  be  re- 
placed by  (2)  or  (3). 

TABLE  VIII. 
VALUES  OF  COEFFICIENT  OF  DISCHARGE  FOR  CIRCULAR  VERTICAL  ORIFICES 


h 
(feet). 

d  (feet). 

0.02 

0.04 

0.07 

0.1 

0.2 

0.6 

1  0 

0.4 

0.637 

0.624 

0.618 

0.6 

0.655 

.630 

.618 

.613 

0.601 

0.593 

0.8 

.648 

.626 

.615 

.610 

.601 

.594 

0.590 

1. 

.644 

.622 

.612 

.608 

.600 

\595 

.591 

1.5 

.637 

.618 

.608 

.605 

.600 

.596 

.593 

2. 

.632 

.614 

.607 

.604 

.599 

.597 

.595 

2. 

.629 

.612 

.605 

.603 

.599 

.598 

.596 

3.5 

.627 

.611 

.604 

.603 

.599 

.598 

.597 

4. 

.623 

.607 

.603 

.602 

.599 

.597 

.596 

6. 

.618 

.607 

.602 

.600 

.598 

.597 

.596 

8. 

.614 

.605 

.601 

.600 

.598 

.596 

.596 

10. 

.611 

.603 

.599 

.598 

.597 

.596 

.595 

20. 

.601 

.599 

.597 

.596 

.596 

.596 

.594 

50. 

.596 

.595 

.594 

.594 

.594 

.594 

.593 

100. 

.593 

.592 

.592 

.592 

.592 

.592 

.592 

*  These  tables  are  taken  from  Hamilton  Smith's  Hydraulics. 


MEASUREMENT  BY  ORIFICES. 


147 


TABLE  IX. 

VALUES  OF  COEFFICIENT  OF  DISCHARGE  FOR  SQUARE  VERTICAL  ORIFICES 


h 
(feet). 

d(feet). 

0.02 

0.04 

0  07 

0   1 

02 

0.6 

1.0 

0.4 

0.643 

0.628 

0.621 

0.6 

0.660 

.636 

.623 

.617 

0.605 

0.598 

0.8 

'  .652 

.631 

.620 

.615 

.605 

.600 

0.597 

1. 

.648 

.628 

.618 

.613 

.605 

.601 

.599 

1.5 

.641 

.622 

.614 

.610 

.605 

.602 

.601 

2. 

.637 

.619 

.613 

.608 

.605 

.604 

.602 

2.5 

.634 

.617 

.610 

.607 

.605 

.604 

.602 

3. 

.632 

.616 

.609 

.607 

.605 

.604 

.603 

5. 

.628 

.614 

.608 

.606 

.605 

.603 

.60^ 

6. 

.623 

.612 

.607 

.605 

.604 

.603 

.602 

8. 

.619 

.610 

.606 

.605 

.604 

.603 

.602 

10. 

.616 

.608 

.605 

.604 

.603 

.602 

.601 

20. 

.606 

.604 

.602 

.602 

.602 

.601 

.600 

50. 

.602 

.601 

.601 

.600 

.599 

.600 

.599 

100. 

.599 

.598 

.598 

.598 

.598 

.598 

.598 

TABLE  X. 

VALUES  OF  COEFFICIENT  OF  DISCHARGE  FOR  RECTANGULAR  ORIFICES 

1  FT.  WIDE. 


ft 

(feet). 

d  (feet). 

0.125 

0.25 

0.50 

0.75 

1.0 

1.5 

2.0 

0.4 
0.6 
0.8 
1. 
1.5 
2. 
2.5 
3. 
4. 
6. 
8. 
10. 
20. 

0.634 

0.633 
.633 
.633 

0.622 
.619. 
.618 
.618 
.618 

0.614 
.612 
.612 
.611 
.611 
.611 

0.608 
.606 
.605 
.605 
.605 
.605 

0.626 
.626 
.624 
.616 
.614 
.612 

0.628 
.630 
.627 
.619 
.616 
.610 

.633 
.633 
.632 
.630 
.629 
.628 
.627 
.624 
.615 
.609 
.606 

.632 
.631 
.620 
.628 
.627 
.624 
.615 
.607 
.603 

.617 
.616 
.615 
.614 
.609 
.603 
.601 

.610 
.609 
.604 
.602 
.601 
.601 

.605 
.602 
.601 
.601 
.601 

.606 
.6Q2 
.601 
.601 

.604 
.602 
.602 

148          THE  MEASUREMENT  OF  RATE  OF  DISCHARGE. 

148.  Miner's  Inch. — Where  water  is  sold  for  hydraulic  min- 
ing or  for  irrigation  the  unit  of  measurement  commonly  em- 
ployed is  the    miner's  inch.      This  is   usually  understood  to 
mean  the  discharge  through  an  orifice  one  inch  square  under 
some  specified  head;  multiples  of  the  "  inch  "  being  obtained  by 
increasing  the  horizontal  dimension  of  the  orifice  while  leaving 
the  head  unchanged.     Various  definitions  have,  however,  been 
accepted   at    different    times   and  in  different  localities.      In 
California     a    common    definition    of    the    miner's    inch    has 
been  the   discharge   from  an   orifice   one   inch  square    when 
the    head    on  the  upper  edge  is  four   inches,   and  hydraulic 
engineers   have   quite  generally  accepted   1.2  cubic    feet   per 
minute  as  its  equivalent,  corresponding  to  a  value  of  about 
0.59  for  c.     The  legal  equivalent*  is,  however,  1.5  cubic  feet 
per  minute. 

149.  Measurement  by  Weirs. — A  weir  is  a  notch  in  the  side 
of  a  vessel  or  reservoir.     Such  notches  are  used  for  the  meas- 
urement of  rate  of  discharge,  and  it  is  essential  that  their  con- 
struction shall  conform  to  accepted  standards  in  order  that 
reliable  results  may  be  obtained. 

The  most  common  form  of  weir  is  rectangular.  The 
standard  rectangular  weir  is  constructed  with  the  lower  edge 
or  " crest"  sharp  (like  the  edge  of  a  standard  orifice),  so 
as  to  permit  full  "crest  contraction"  of  the  stream.  The 
vertical  edges  should  either  be  sharp  or  else  should  lie  in 
the  vertical  sides  of  the  channel  of  approach.  In  the 
former  case  there  is  full  "end  contraction"  of  the  stream; 
in  the  latter  there  will  be  no  lateral  contraction  at  the 
ends,  which  is  commonly  expressed  by  saying  that  the  end 
contractions  are  "suppressed." 

It  is  essential  also  that  the  depth  of  the  channel  of  approach 
shall  be  sufficient  so  that  full  crest  contraction  is  not  interfered 
with.  Also,  unless  the  end  contractions  are  completely  sup- 
pressed, the  channel  of  approach  should  be  considerably  wider 
than  the  length  of  the  weir,  in  order  that  full  contraction  may 
occur  at  the  ends. 

*  *  Since  1901. 


THE   FRANCIS  WEIR  FORMULA.  149 

The  formula  for  the  rate  of  discharge  over  a  rectangular 
weir  is  given  in  Art.  50.     It  is 


c  being  a  coefficient  whose  value  must  be 
determined  by  experiment. 

The  "head  on  the  crest"  (H)  must  be 
measured  to  the  level  of  the  surface  of  still  FIG.  74. 

water  back  of  the  weir  (Fig.  74). 

150.  Thfc  Francis  Formula. — Elaborate  experiments  were 
made  by  J.  B.  Francis  *  for  the  purpose  of  determining  the 
value  of  the  coefficient  c  in  the  above  formula,  and  also  the 
effect  of  end  contractions.  The  weirs  used  in  these  investiga- 
tions ranged  in  length  from  4  ft.  to  10  ft.,  and  H  ranged  from 
0.6  ft.  to  1.6  ft.  As  a  result  of  these  experiments  their  author 
adopted  an  average  value  0.622  for  the  coefficient  c.  His 
formula,  for  the  case  of  suppressed  end  contractions,  is 


(6) 


b  and  H  being  in  feet. 

As  regards  end  contractions,  Francis  concluded  that  each 
contraction  decreases  the  effective  length  of  the  weir  by  an 
amount  proportional  to  H,  and  gave  the  formula 


(7) 


in  which  n  is  the  number  of  end  contractions,  usually  two. 

Velocity  of  approach.  —  If  the  channel  of  approach  is  so  small 
that  the  velocity  of  the  approaching  water  is  appreciable,  the 
formula  of  Art.  68  is  taken  as  the  basis,  giving  the  following: 

q  =  3.33(b-Q.lnH)[(H  +  h')3*-h'i],        ...     (8) 

i/2 
in  which  h'  =  -^-,  v'  being  the  velocity  of  approach.     In  prac- 

Z9 

tice  !/  is  taken  as  the  average  velocity  in  the  channel,  computed 
from  the  cross-section  and  the  rate  of  discharge. 
*  Lowell  Hydraulic  Experiments. 


150 


THE  MEASUREMENT   OF   RATE   OF  DISCHARGE. 


It  is  probable  that  for  small  values  of  H  the  constant  co- 
efficient adopted  by  Francis  gives  results  a  little  too  great.  In 
practice,  however,  cases  are  rare  in  which  measurements  can 
be  made  with  a  degree  of  reliability  warranting  the  use  of 
formulas  or  coefficients  claiming  greater  accuracy  than  those 
of  Francis.  Fanning  *  recommends,  for  use  in  the  formula  of 
Francis,  values  of  the  coefficient  varying  with  H  as  in  the 
following  table : 

TABLE  XI. 


H 

(feet). 

c 

c§v/20 

// 

(feet). 

c 

ciVjfc 

.083 

.6100 

3.263 

.833 

.6240 

3.338 

.125 

.6120 

3.274 

1.000 

.6241 

3.339 

.167 

.6140 

3.285 

1.167 

.6242 

3.339 

.250 

.6170 

3.301 

1.333 

.6243 

3.340 

.333 

.6195 

3.314 

1.500 

.6242 

3.339 

.500 

.6223 

3.329 

1.667 

.6241 

3.339 

.667 

.6235 

3.336 

2.000 

.6240 

3.338 

151.  Method  of  Hamilton  Smith. — The  formula  adopted  by 
Smith  differs  from  that  of  Francis  in  the  method  of  allowing 
for  end  contractions  and  also  in  the  correction  for  velocity  of 
approach.  Smith  also  adopts  values  of  c  varying  with  the 
head,  even  when  the  end  contractions  are  suppressed.  The 
formula 


(9) 


is  used  for  both  contracted  weirs, and  weirs  with  end  contrac- 
tions suppressed,  but  the  values  of  c  are  different  for  the 
two  cases,  as  shown  in  Tables  XII  and  XIII. 

Velocity  of  approach. — If  the  velocity  of  approach  is  not 
negligible,  let  h'  denote  the  head  equivalent  to  the  mean  veloc- 
ity in  the  channel  above  the  weir,  and  substitute  for  H  in 
equation  (9) 

for  weir  with  end  contractions  suppressed, 
Ah'   "     "       "     full  end  contractions. 


*  A  Practical  Treatise  on  Hydraulic  and  Water-Supply  Engineering. 


WASTE   WEIRS. 


151 


TABLE  XII. 

VALUES  OF  THE  COEFFICIENT  OF  DISCHARGE  (c)  FOR  WEIRS  WITHOUT 
END  CONTRACTIONS. 


H 

(feet). 

Length  (6)  in  Feet. 

0.66 

2 

3 

4 

5 

7 

10 

19 

0.1 

0.675 

0.659 

0.658 

0.658 

0.675 

0.15 

.662 

0.652 

0.649 

0.647 

.645 

.645 

.644 

.643 

0.2 

.656 

.645 

.642 

.641 

.638 

.637 

.637 

.635 

0.2  5 

.653 

.641 

.638 

.636 

.634 

.633 

.632 

.630 

0.3 

.651 

.639 

.636 

.633 

.631 

.629 

.628 

.626 

0.4 

.650 

.636 

.633 

.630 

.628 

.625 

.623 

.621 

0.5 

.650 

.637 

.633 

.630 

.627 

.624 

.621 

.619 

0.6 

.651 

.638 

.634 

.630 

.627 

.623 

.620 

.618 

0.7 

.658 

.640 

.635 

.631 

.628 

.624 

.620 

.618 

0.8 

.656 

.643 

.637 

"  .633 

.629 

.625 

.621 

.618 

0.9 

.645 

.639 

.635 

.631 

.627 

.622 

.619 

1.0 

.648 

.641 

.637 

.633 

.628 

.624 

.619 

1.2 

.646 

.641 

.636 

.632 

.626 

.620 

1.4 

.644 

.640 

.634 

.629 

.622 

1.6 

.647 

.642 

.637 

.631 

.623 

TABLE  XIII. 

VALUES  OF  COEFFICIENT  OF  DISCHARGE  (c)  FOR  WEIRS  WITH 
END  CONTRACTIONS. 


H 

(feet). 

Length  (6)  in  Feet. 

0.66 

l 

2 

3 

5 

10 

19 

0.1 

0.632 

0.639 

0.643 

0.652 

0.653 

0.655 

0.656 

0.15 

.619 

.625 

.634 

.638 

.640 

.641 

.642 

0.2 

.611 

.618 

.626 

.630 

.631 

.633 

.634 

0.25 

.605 

.612 

.621 

.624 

.626 

.628 

.629 

0.3 

.601 

.608 

.616 

.619 

.621 

.624 

.625 

0.4 

.595 

.601 

.609 

.613 

.615 

.618 

.620 

0.5 

.590 

.596 

.605 

.608 

.611 

.615 

.617 

0.6 

.587 

.593 

.601 

.605 

.608 

.613 

.615 

0.7 

.590 

.598 

.603 

.606 

.612 

.614 

0.8 

.595 

.600 

.604 

.611 

.613 

0.9 

.592 

.598 

.603 

.609 

.612 

1.0 

.590 

.595 

.601 

.608 

.611 

1.2 

.585 

.591 

.597 

.605 

.610 

1.4 

.580 

.587 

.594 

.602 

.609 

1.6 

.582 

.591 

.600 

.607 

152.  Waste  Weirs. — A  waste  weir  is  a  notch  left  at  the 
crest  of  a  dam  for  the  purpose  of  discharging  flood  water  with- 
out injury  to  the  dam. 

To  choose  the  proper  length  and  depth  of  the  waste  weir, 


152         THE  MEASUREMENT  OF  RATE  OF  DISCHARGE. 

knowing  the  greatest  rate  of  accumulation  of  flood  water,  the 
general  weir  formula  may  be  used.  Since  a  waste  weir  will  not 
in  general  have  a  sharp  crest,  and  since  the  other  conditions 
are  not  the  same  as  in  the  case  of  weirs  constructed  for  the 
purpose  of  accurate  measurement  of  rate  of  discharge,  no  pre- 
cise estimate  can  be  made  of  the  rate  of  discharge  of  a  waste- 
weir  under  a  given  head. 

Francis  gave  the  empirical  formula 


(10) 


in  which  b  and  H  are  in  feet,  and  q  is  in  cubic  feet  per  second. 

The  ordinary  weir  formula,  with  the  coefficient  determined 
by  Francis,  may  be  used  with  results  differing  little  from  those 
given  by  equation  (10).  That  is, 


(11) 


153.  Triangular  Weir.  —  Although  the  rectangular  form  of 
weir  is  generally  the  most  convenient,  a  triangular  notch  may 
be  used  when  the  quantity  to  be  measured  is  not  too  great. 
Theoretically  the  triangular  form  has  the  advantage  that  the 
shape  of  the  water  section  does  not  change  with  the  head,  so 
that  the  coefficient  of  discharge  would  be  expected  to  be  more 
nearly  constant  with  varying  head  than  in  the  case  of  the  rect- 
angular form.  This  is  confirmed  by  experiment. 

The  formula  for  the  rate  of  discharge  over  a  triangular  notch, 
as  deduced  in  Art.  51,  is 


.    ,     ;     .     .     ,     (12) 
If  the  angle  at  the  vertex  is  90°,  6  =  2H,  and  the  formula  becomes 

(13) 


If  the  edges  are  made  sharp,  as  in  the  case  of  the  standard  rect- 
angular weir,  the  value  of  c  for  heads  under  0.8  ft.  may  be 


SUBMERGED  WEIR.  153 

taken  as  .592.     With  the  foot  as  unit  length  the  formula  may 
be  written 

(14) 


EXAMPLES. 

1.  Compute  the  rate  of  discharge  of  an  orifice  6"  square  when  the 
head  on  the  center  is  18  ft.  Ans.  5.12  cu.  ft.  per  sec. 

2.  Estimate  the  diameter  of  a  circular  orifice  to  discharge  1  cu.  ft. 
per  sec.,  the  head  on  the  center  being  5  ft.  Ans.  0.345  ft. 

3.  A  weir  without  end  contractions  is  4  ft.  long,  the  bottom  of  the 
channel  of  approach  being  3  ft.  below  the  crest.     Compute  the  rate  of 
discharge  when  H  =0.75  ft.  (a)  by  formula  of  Francis  and  (6)  by  formula 
of  Smith.     What  percentage  of  error  is  caused  by  neglecting  velocity  of 
approach?  Ans.  (a)  8.73  cu.  ft.  per  sec.     (6)  8.90  cu.  ft.  per  sec. 

4.  A  contracted  weir  9  ft.  long  is  to  discharge  60  cu.  ft.  per  sec.     Esti- 
mate the  head  on  the  crest  (a)  by  Francis'  formula  and  (6)  by  Smith's 
formula.  Ans.  (a)  H  =1.630  ft.     (6)  H  =  1.572  ft. 

5.  Water  enters  a  reservoir  at  the  rate  of  20  cu.  ft.  per  sec.     In  the 
side  is  a  rectangular  notch  3.5  ft.  long,  with  ends  and  crest  so  formed 
as  to  allow  complete  contraction.     Estimate  the  head  on  the  crest  when 
a  steady  condition  is  attained.  Ans.  H  =  1  .52  ft. 

6.  A  rectangular  orifice  1  ft.  wide  and  4  in.  deep  is  formed  in  the 
vertical  side  of  a  reservoir  into  which  water  flows  at  the  rate  of  5  cu.  ft. 
per  sec.     How  high  above  the  center  of  the  orifice  will  the  water  surface 
rise?  Ans.  About  9.6  ft. 

7.  Compute  the  rate  of  discharge  of  a  triangular  notch  having  a 
vertex  angle  of  90°  when  the  head  is  0.46  ft.     Ans.  0.365  cu.  ft.  per  sec. 

154.  Submerged  Weir.  —  A  weir  is  said  to  be  submerged  if 
the  surface  of  the  stream  below  is  higher  than  the  crest  of  the 
weir.  To  deduce  a  formula  for  the  rate  of  discharge  in  such  a 
case,  let  H,  h  denote  the  elevations  of  the  up-stream  and  down- 
stream surfaces  respectively  above  the  crest,  and  b  the  hori- 
zontal length  of  the  weir  (Fig.  75).  The  total  discharge  orifice 
ABFE  consists  of  two  parts,  ABDC,  CDFE,  of  which  the  former 
may  be  treated  as  an  ordinary  weir  and  the  latter  as  a  sub- 
merged orifice.  Omitting  coefficients  of  discharge,  the  formula 
for  the  ordinary  weir  ABDC  is 


154          THE   MEASUREMENT  OF  RATE  OF  DISCHARGE, 
and  that  for  the  submerged  orifice  CDFE  is 


q  =  bhV2g(H-h). 

Combining  these,  and  introducing  the  coefficient  of  discharge  c, 
the  formula  for  the  rate  of  discharge  of  the  submerged  weir 
becomes 


(15) 


, _A| B 

i  \_ ,_     ._CL_  _D 

x_-x*^ 

^ 
" *       B 6_Y.v 

L I 

FIG.  75. 

Equivalent  .ordinary  weir. — If  Hr  is  the  head  on  an  unsub- 
merged  weir  whose  rate  of  discharge  is  q,  we  have 

(16) 

Assuming  the  coefficients  of  discharge  for  the  two  cases  to  be 
equal,  the  values  of  q  given  by  (15)  and  (16)  will  be  equal  if 

or 

Writing  n  for  H' /H,  and  assuming  c  to  have  the  mean  value 
found  by  Francis  for  ordinary  weirs,  trie  formula  for  the  rate  of 
discharge  over  a  submerged  weir  without  end  contractions  may 
be  written 

(18) 


in  which  n  is  to  be  computed  from  (17). 


SUBMERGED  DAM. 


155 


The  relation  between  h/H  and  n  as  given  by  equation  (17) 

is  shown  by  the  curve  (B)  in  Fig.  76.     Curve  (A)  shows  the 

.2  .4  __.G  .8  1.0 


,(Bj 


\ 


\ 


.4 


.2 


.8 


1.0 


FIG.  76. 


relation  deduced  by  Herschel  *  from  experiments  made  by 
Francis  and  by  Ftely  and  Stearns.  These  results  are  also  given 
in  Table  XIV. 

155.  Submerged  Dam. — If  the  channel  of  a  stream  is  ob- 
structed by  a  dam,  the  crest  of  which  is  lower  than  the  original 
surface  of  the  stream,  the  surface  up-stream  from  the  obstruc- 
tion will  be  raised  an  amount  depending  upon  the  position  of 
the  crest,  the  length  of  the  overfall,  and  the  rate  of  discharge. 
An  approximate  estimate  of  the  effect  may  be  made  by  the 
formula  above  given  for  submerged  weirs. 

*  Transactions  American  Society  of  Civil  Engineers,  Vol.  XIV,  p.  189. 


156          THE  MEASUREMENT  OF  RATE   OF  DISCHARGE. 

TABLE  XIV. 


n 

n 

n 

h 

h 

h 

H 

Theoreti- 

Experi- 

H 

Theoreti- 

Experi- 

H 

Theoreti- 

Experi- 

cal. 

mental. 

cal. 

mental. 

cal. 

mental. 

0 

1.000 

1.000 

.45 

.959 

.912 

.84 

.724 

.656 

.05 

1.000 

1.007 

.50 

.945 

.892 

.86 

.697 

.631 

.10 

.999 

1.005 

.55 

.928 

.871 

.88 

.666 

.604 

.15 

.997 

.996 

.60 

.908 

.846 

.90 

.631 

.574 

.20 

.995 

.985 

.65 

.883 

.818 

.92 

.589 

.539 

.25 

.991 

.972 

.70 

.853 

.787 

.94 

.538 

.498 

.30 

.986 

.959 

.75 

.816 

.750 

.96 

.473 

.441 

.35 

.979 

.944 

.80 

.770 

.703 

.98 

.378 

.352 

.40 

.970 

.929 

.82 

.748 

.681 

1.00 

.000 

.000 

As  an  example,  suppose  a  stream  discharging  600  cubic  feet 
per  second  is  obstructed  by  a  dam  with  an  overfall  40  feet  long, 
the  crest  being  2  feet  below  the  original  water  surface.  The 
increase  in  depth  above  the  dam  may  be  estimated  as  follows: 

With  the  notation  of  Art.  154,  the  value  of  H'  '  ,  the  head 
on  the  equivalent  weir,  is  found  from  the  equation 

600  =  3.33  X40#'s, 


or 


Also,  since  h  =  2  ft., 


nH    2.728 


1.364, 


or 


1.364^. 


By  trial  substitutions  in  the  table,  using  the  experimental  values 
of  n,  it  is  found  that  h/H  =  .6l5  very  nearly,  from  which  H  = 
3.25  ft. 

If  the  curves  in  Fig.  76  be  carefully  drawn  on  cross-section 
paper,  the  above  solution  may  be  shortened.    The  equation 

n- 1.36477  represents  a  straight  line  whose  intersection  with 
curve  (A)  determines  n  and  h/H  at  once. 


VENTURI  METER.  157 

As  a  second  example,  let  it  be  required  to  determine  where 
the  crest  of  the  dam  should  be  placed  in  order  to  raise  the  sur- 
face 1  foot,  the  data  being  otherwise  as  above  given.  From  the 
condition  stated, 


and  as  before 

H'  =  nH  =  2.728  ft., 

2.728 


or 


H  = 


n 
From  these  equations, 


H    2.728' 

By  trial  substitution  in  the  table  it  is  found  that  n  =  .775,  from 
which  #  =  3.52  ft.  and  h  =  2.52  ft. 

This  case  may  also  be  solved  graphically,  by  finding  the  in- 

h  Ti 

tersection  of  the  straight  line  1  —  77  =  0  70Q  with  curve  (A)  in 

n      Z./Zo 


Fig.  76. 


EXAMPLES. 


1.  A  stream  discharging  550  ou.  ft.  per  sec.  is  obstructed  by  a  dam 
of  which  the  crest  is  2.5  ft.  below  the  original  water  surface,  the  length 
of  the  overfall  being  50  ft.     What  will  be  the  effect  on  the  water  surface 
above  the  dam?  Ans.  H-h=O.Q3  ft. 

2.  Where  must  the  crest  of  a  dam  be  placed  in  order  that  the  water 
may  be  raised  2  ft.,  the  length  of  the  overfall  being  30  ft.  and  the  rate 
of  discharge  500  cu.  ft.  per  sec.?  Ans.  h  =  1.10  ft. 

156.  Venturi  Meter.  —  The  essential  features  of  the  Venturi 
meter  have  already  been  shown  in  Figs.  35  and  36,  illustrating 
examples  given  at  the  end  of  Chapter  V.  The  instrument  con- 
sists merely  of  a  converging  tube  with  piezometers  connected 
substantially  as  shown  in  Fig.  77,  or  a  difference-gauge,  as  in 
Fig.  78. 


158 


THE   MEASUREMENT   OF  RATE   OF  DISCHARGE. 


Referring  first  to  the  arrangement  in  Fig.  77,  let  the  usual 
notation  be  used  for  cross-sectional  areas,  velocities,  and  heights 
of  piezometer  columns,  and  let  the  equation  of  energy  be  written, 
A  being  the  up-stream  and  B  the  down-stream  section.  Then, 


A' 

T~ 

b 

B' 

^ 

A                         B 
FIG.  77. 

A" 

—  i  — 

B' 

li 

L 

A' 

M 

C 

B 

FIG.  78. 


whether  the  axis  of  the  pipe  be  horizontal  or  not,  we  have,  as 
in  Art.  125, 


H'  being  the  head  lost  between  A  and  B,  and  ki,  k2  being  posi- 
tive quantities  whose  values  depend  upon  the  distribution  of 
velocities  throughout  the  cross-sections.  Neglecting  ki,  k2  and 
H',  and  introducing  the  relation  FiVi=F2v2,  there  results  the 
equation 


in  which  c  is  a  coefficient  whose  value  would  be  1  if  the  theory 
were  perfect,  and  whose  value  as  found  by  experiment  usually 
ranges  from  1  to  0.97.  The  value  of  c  decreases  with  increasing 
velocity. 

The  factor  y\—y^  in  the  above  formula  is  the  difference  in 
level  between  the  tops  of  the  two  piezometer  columns  Af  and 
B'  (Fig.  77).  If  the  pressure  is  too  great  for  the  use  of  open 
piezometers,  a  difference-gauge  may  be  connected  as  in  Fig.  78. 
If  the  specific  gravity  of  the  mercury  or  other  fluid  in  the  U 
tube  is  s,  and  if  h  is  the  difference  in  level  of  the  two  columns, 


VENTURI  METER.  159 

as  indicated  in  Fig.  78,  the  value  of  yi  -y2  in  the  formula  for 
v2  is  (s-l)h.     For  mercury  the  value  of  s  is  very  nearly  13.6. 

EXAMPLES. 

1.  A  Venturi  meter  whose  two   diameters  were  48  inches  and  16 
nches  was  furnished  with  piezometers  as  in  Fig.  77.     The  difference  in 
level  between  the  tops  of  the  two  columns  was  found  to  be  1.27  ft.  in  a 
certain  case.     Compute  the  rate  of  discharge,  assuming  c  =0.985. 

Ans.  12.5  cu.  ft.  per  sec. 

2.  A  Venturi  meter  of  the  dimensions  given  in  Ex.  1  was  furnished 
with  a  mercury  difference-gauge,  as  in  Fig.  78.     In  a  certain  experiment 
the  difference  in  elevation  of  the  two  mercury  columns  in  the  U  tube 
was  0.625  ft.     Estimate  the  rate  of  discharge,  taking  c  =0.98. 

Ans.  31.0  cu.  ft.  per  sec. 


CHAPTER   XIV. 
DYNAMIC   ACTION    OF   STREAMS. 

157.  Meaning  of  Dynamic  Action.  —  The  object  of  the  pres- 
ent chapter  is  to  investigate  the  forces  which  are  exerted  by  a 
stream  upon  bodies  which  constrain  its  motion,  and  the  reac- 
tions exerted  by  these  bodies  upon  the  stream. 

Thus,  if  a  free  jet  of  water  is  deflected,  forces  are  called  into 
action  between  the  particles  of  the  jet  and  the  body  causing 
the  deflection.  The  same  is  true  in  the  case  of  a  confined 
stream  whose  velocity  varies  in  magnitude  from  section  to 
section  because  the  cross-section  of  the  pipe  changes,  or  in 
direction  because  the  pipe  bends.  These  are  illustrations  of 
dynamic  action  in  cases  of  steady  flow. 

Another  important  case  is  that  in  which  the  rate  of  dis- 
charge of  a  confined  stream,  and  therefore  the  velocity  at  every 
section,  are  suddenly  changed,  as  by  the  closing  of  a  valve. 

In  all  cases,  dynamic  effects  are  to  be  determined  by  apply- 
ing the  fundamental  laws  of  motion  or  the  equations  based 
upon  them. 

158.  Principle  of  Momentum.  —  If  a  constant  force  P  acts 
upon  a  body  of  mass  m  for  a  time  M,  thereby  giving  its  veloc- 
ity *  the  increment  4V,  we  have 


« 


*  Throughout  the  remainder  of  this  book  it  is  necessary  to  distinguish 
between  absolute  and  relative  velocities  of  a  stream.  We  therefore  use  V 
for  the  former  and  v  for  the  latter.  See  Appendix  B  for  a  discussion  of 
absolute  and  relative  motion. 

160 


FORCE  PRODUCING  VELOCITY  OF  JET  FROM   ORIFICE.    161 

If  the  force  is  not  constant,  the  equation  gives  its  average 
value  during  the  time  At. 

Thus  if  a  body  of  12  Ibs.  mass  (m  =  12/0  =  .373  if  force  is 
to  be  expressed  in  pounds)  has  its  velocity  changed  in  .1  sec. 
from  18  ft.  per  second  in  a  given  direction  to  23  ft.  per  second 
in  the  same  direction,  the  average  value  of  the  force  during 
that  time  is 


_ 

If  At  =  l  sec.,  the  equation  is 

(2) 


That  is,  the  change  of  momentum  produced  by  a  force  in  one 
second  is  equal  to  the  value  of  the  force  if  constant,  or  to  its 
average  value  for  that  second  if  variable. 

159.  Force  Producing  Velocity    of   Jet   from    Orifice.  —  Let 

W  Ibs.  of  water  per  second  flow  from  an  orifice  in  the  side  of  a 
reservoir,  V  being  the  velocity  of  the  jet.  The  velocity  of  each 
particle  is  then  changed  by  the  amount  V  during  its  passage 
from  the  reservoir  to  the  smallest  cross-section  of  the  jet.  The 
force  causing  this  change  is  exerted  directly  by  contiguous 
particles,  but  indirectly  by  the  walls  of  the  reservoir.  The 
values  of  the  forces  exerted  upon  individual  particles  cannot  be 
determined,  but  the  sum  of  the  average  forces  for  all  particles 
can  be  computed  from  the  principle  of  momentum  above  stated. 
In  one  second  the  mass  of  water  flowing  from  the  orifice  is 
TF/0,  and  the  total  momentum  produced  in  one  second  by  the 
action  of  the  reservoir  upon  the  water  is  WV/g.  This  is  there- 
fore the  average  value  of  the  total  force  continually  exerted  by 
the  reservoir  upon  the  stream.  This  force  is  practically  con- 
stant since  the  flow  is  steady,  hence  its  average  value  is  its 
actual  value.  If  F  is  the  cross-section  of  the  jet,  W  =  wFVf 
and  the  force  is 


162 


DYNAMIC  ACTION  OF  STREAMS. 


160.  Reaction  of  Jet  upon  Reservoir.  —  By  the  law  of  action 
and  reaction,  the  particles  of  the  jet  exert  upon  the  reservoir 
forces  whose  resultant  is  equal  and  opposite  to  the  force  just 
computed. 

EXAMPLES. 

1.  Determine  the  reaction  upon  the  reservoir  due  to  a  jet  from  a 
standard  circular  orifice  I"  in  diameter  under  a  head  of  3'. 

Ans.  1.21  Ibs. 

2.  Show  that  if  there  were  no  loss  of  energy,  the  reactive  force  exerted 
by  a  jet  from  an  orifice  would  be  double  the  total  static  pressure  upon 
an  area  equal  to  the  cross-section  of  the  jet,  due  to  the  head  on  the 
orifice. 

161.  Jet   Striking  a  Fixed   Surface  Normally.  —  If   a  jet  is 
intercepted  by  a  plane  surface  perpendicular  to  the  axis  of  the 

jet,  the  resultant  action  and  reaction  between  the 
stream  and  the  body  will  be  directed  normally  to 
the  surface.  If  V  is  the  velocity  of  the  jet,  the 
increment  of  velocity  for  each  particle,  resolved 
in  the  direction  of  the  normal,  is  —  7,  so  that 
the  resultant  force  exerted  by  the  body  upon  the 
jet  is 

pJ^p^.^T.    .    .     (4) 

in  the  direction  opposite  to  the  motion  of  the  jet.    The  reaction 
of  the  jet  upon  the  fixed  body  is  equal  and  opposite  to  this. 


FIG.  79. 


Jet  Striking  a  Moving  Surface  Normally.  —  Let  a  jet 
whose  velocity  is  V  strike  normally  against  the  surface  of  a 
body  which  is  itself  moving  with  velocity 
u  in  the  same  direction  as  the  jet.  The 
increment  of  velocity  for  each  particle, 
resolved  normally  to  the  surface,  has  the 
magnitude  V  —  u.  If  W  denotes  the 
number  of  pounds  of  water  striking  the 
surface  per  second,  the  force  exerted  upon 
the  moving  body  has  the  value 

W 


JET  WATER  WHEEL  WITH  FLAT  VANES.  163 

Or,  since  W'  =  wF(V-u), 

P'^-(V-u?  .......     (5) 

Work  done  upon  moving  body.  —  The  work  done  in  one  sec- 
ond by  the  force  P'  is 

P'u  =  ^-u(V-u)2  .......     (6) 

t7 

163.  Jet  Water  Wheel  with  Flat  Vanes.  —  Fig.  81  repre- 
sents a  wheel  provided  with  flat  vanes  against  which  a  jet  of 
water  is  directed.  The  results  of  Art.  162 

• 

would  be  strictly  applicable  if  the  vanes 

always    received    the    jet    normally.    Al- 

though this  condition  cannot  be  realized, 

the  above  results  may  be  used  as  an  ap- 

proximation,  and  equations   (5)   and  (6) 

may  be  employed  to   compute   the  force 

exerted  upon  a  vane,  and  the  work  done 

by  this  force  per  seqond.    The  total  action  of  the  jet  upon  the 

wheel  is  not,  however,  the  same  as  its  action  upon  one  vane, 

since  more  than  one  vane  will  be  receiving  the  jet  at  the  same 

time. 

Replacing  W  by  W  ,  the  weight  of  water  discharged  by  the 
jet  per  second,  we  get  for  the  force  exerted  upon  the  wheel 


(7) 


and  for  the  work  done  upon  the  wheel  per  second 


(8) 


Maximum  value  of  work  done  upon  wheel.  —  If  u  varies,  the 
value  of  the  work  varies,  having  its  maximum  when  u  =  V/2, 
which  gives 

wFV3    WV2 
Maximum  £  =  —£--  =  =  —  ......     (9) 


164 


DYNAMIC  ACTION  OF  STREAMS. 


Since  the  kinetic  energy  of  W  Ibs.  of  water  in  the  jet  is  WV2/2g, 
it  is  seen  that  not  more  than  half  of  this  can  be  utilized  by  a 
water  wheel  with  flat  vanes. 

164.  Force  Causing  Deflection  of  a  Jet. — If  the  velocity  of 
a  particle  changes  in  direction,  the  average  value  of  the  force 
acting  upon  it  during  a  time  Jt  is  still  given  by  the  formula 


mAV 

At  ' 


(10) 


B 


but  AV  is  now  the  vector  increment  *  of  velocity. 

In  Fig.  82  is  represented  a  jet  of  water  which  is  deflected 
by  a  curved  vane  MN,  the  velocity  of  every  particle  being 
changed  from  Vi  (represented  by  the  vector  OA)  to  V%  (repre- 
sented by  the  vector  OB).  The  increment  of  velocity  is  then  rep- 
resented by  the  vector  AB.  If  the  discharge  of  the  jet  is  W 

Ibs.  per  second,  the  change  of 
momentum  produced  in  one  sec- 
ond by  the  action  of  the  deflect- 
ing surface  is 


W 

—JV, 

9      ' 


which  is  therefore  the  value  of 
the  resultant  force  constantly 
exerted  upon  the  jet  by  the  deflecting  body.  If  Vlt  V2  and 
the  angle  of  the  deflection  a  are  known,  the  magnitude  and 
direction  of  AV  can  be  computed  by  solving  the  triangle  OAB. 
The  line  of  action  of  the  resultant  force  passes  through  the 
intersection  of  the  two  lines  which  coincide  with  the  axis  of  the 
jet  before  and  after  the  deflection,  as  indicated  in  Fig.  82. 
If  the  magnitude  of  the  velocity  is  not  changed,  so  that 


FIG 


*  Theoretical  Mechanics,  Art.  253. 


VELOCITY  OF  STREAM  REVERSED.  165 

W  ...    2WV        a    2wFV2   .    a 
and  p=-jy  =  —  sm2=  —  sm^,       .     .     (11) 

a  being  the  angle  of  deflection,  i.e.,  the  angle  between  Vl  and  V2. 
In  any  case,  the  reaction  of  the  jet  upon  the  body  which 
deflects  it  is  equal  and  opposite  to  the  force  exerted  upon  the 
jet. 

EXAMPLES. 

1.  A  jet  of  2  sq.  in.  cross-section,  having  a  velocity  of  40  ft.  per  sec., 
strikes  a  curved  surface  which  deflects  it  25°  and  changes  its  velocity 
to  35  ft.  per  sec.     Determine  the  magnitude  and  direction  of  the  force 
exerted  by  the  jet  upon  the  deflecting  body. 

Ans.  A  force  of  9.15  Ibs.  making  angle  of  60°  46'  with  jet. 

2.  If  the  jet  in  Ex.  1  is  deflected  25°  without  diminution  of  velocity, 
determine  the  force  exerted  upon  the  deflecting  body. 

Ans.  9.34  Ibs.  making  angle  of  77°  30'  with  jet. 

165.  Case  in  which  the  Velocity  of  the  Stream  is  Reversed. 

—  If  the  deflection  of  the  stream  amounts  to  180°,  as  in  Fig. 
33,  the  increment  of  velocity  is  Vi  +  V2,  and  the  force  exerted 
by  the  deflecting  body  is 


W  * 

P=JV1+y2)      IVM+VJ,    .  .  .   (12) 


if  FI  is  the  cross-section  of  the  stream  where  its  velocity  is  V 


FIG.  83.  FIG.  84. 

If  the  jet  could  be  reversed  without  diminution  of  its  velocity 
(Fig.  84),  so  that  V2  =  Vi  =  V,  the  force  would  have  the  value 


The  line  of  action  of  P  is  found  from  the  law  of  composition 
of  parallel  forces.    The  momentum  WV2/g  is  the  resultant  of  the 


166  DYNAMIC  ACTION   OF  STREAMS. 

momentum  WVi/g  and  that  due  to  the  force  P  acting  for  cne 
second.  The  distance  between  the  lines  lying  in  the  axis  of  the 
jet  before  and  after  deflection  is  therefore  divided  by  the  line 
of  action  of  P  in  the  inverse  ratio  of  V\  and  V2. 

166.  Reversal   of  Jet  by  Moving  Vane.  —  In  the  preceding 
case  suppose  the  curved  vane  which  deflects    the  jet  to  be 

moving  with  velocity  u  in  the  same 
direction  as  the  impinging  jet.  If 
V\  and  V2  are  the  values  of  the 
absolute  velocity  of  the  stream  just 
before  striking  the  vane  and  just 

a^ter  ^eavm§  ^>  anc^  if  ^7/  denotes 
the  weight   of  water   striking  the 
vane  per  second,  the  force  exerted  by  the  jet  upon  the  vane  is 


The  values  of  V2  and  of  W  will  depend  upon  u. 

Let  v  denote  the  velocity  of  a  particle  relative  to  the  vane  at 
a  point  where  its  absolute  *  velocity  is  V  ;  v\  being  the  value  of 
v  at  the  point  where  the  stream  is  about  to  strike  the  vane, 
and  v2  its  value  at  the  point  where  the  stream  leaves  the  vane, 
Then 


If  there  were  no  loss  of  energy  in  the  flow  over  the  vane,  v2 
would  equal  vi.    Assuming  this  to  be  true, 


Substituting  in  the  above  value  of  P',  and  noticing  that 


we  have  P-(Fi-w)*  .......     (14) 

y 

*  See  Appendix  B. 


JET  WATER   WHEEL   WITH  CURVED  VANES.  167 

Work  done  on  moving  vane.  —  The  work  done  per  second  by 
the  force  P*  is 


(15) 


Comparing  with  Art.  162,  it  is  seen  that  these  values  of  the 
force  and  the  work  are  double  those  found  in  the  case  of  a  flat 
vane. 

167.  Jet  Water  Wheel  with  Curved  Vanes.  —  If  the  flat 
vanes  in  Fig.  81  were  replaced  by  curved  vanes  so  arranged  as 
to  receive  and  discharge  the  jet  substantially  as  in  Fig.  85,  the 
above  reasoning  would  hold  approximately,  equations  (14)  and 
(15)  representing  the  force  exerted  upon  a  single  vane  and  the 
work  done  by  that  force  per  second.  Since,  however,  more 
than  one  jet  would  be  receiving  water  at  the  same  time,  so  that 
the  wheel  would  be  acted  upon  by  all  the  water  discharged  by 
the  jet,  W  must  be  replaced  by  W  to  get  the  total  action  upon 
the  wheel,  in  which 


That  is,  the  total  force  exerted  upon  the  wheel  would  be 


P-y(F,  +  F2)  -F,(Fi  +  Fa)  --Fi(F,  -u)  ;  (16) 
y  y  y 

and  the  work  done  upon  the  wheel  per  second  would  be 

1(71-u)w  .....     (17) 
y 

These  values  are  double  those  obtained  in  Art.  163  for  a  wheel 
with  flat  vanes. 

The  maximum  value  of  the  work,  given  by  u  =  Vi/2,  is 

WV}2 
Maximum  L  =  —  ^  —  ,      .....     (18) 

J/ 

which  is  equal  to  the  whole  kinetic  energy  of  the  jet. 

This  form  of  water  wheel  is  discussed  in  Chapter  XVIII. 


168  DYNAMIC  ACTION  OF  STREAMS. 

,  168.  Principle  of  Angular  Momentum. — From  the  principle 
stated  in  Art.  158  another  of  equal  importance  may  be  deduced. 

Since  the  total  increment  of  momentum  of  a  particle  per 
second,  is  equal  to  the  average  value  of  the  force  acting  upon 
it,  it  follows  by  taking  moments  about  any  axis  that  the  total 
increment  of  the  moment  of  momentum  (or  "angular  momen- 
tum") per  second  is  equal  to  the  average  value  of  the  moment 
of  the  force. 

This  principle  is  of  use  in  the  following  discussion. 

169.  Action  of  Stream  upon  Rotating  Vane  in  General  Case. 
—All  the  special  cases  above  considered  are   covered  by  the 
following  general  discussion. 

Let  MN  (Fig.  86)  represent  a  curved 
vane  which  rotates  uniformly,  with  angu- 
lar velocity  cu,  about  an  axis  at  0  per- 
pendicular to  the  plane  of  the  figure. 
Let  the  vane  receive  a  stream  at  M  and 
discharge  it  at  N.  The  point  M  of  the 
vane  moves  at  every  instant  at  right 
angles  to  OM  with  velocity  HI,  and 
/  the  point  N  moves  perpendicularly  to 

FIG.  86.  ON  with  velocity  u^. 

Let  Vi,  Vi=  absolute  and  relative  *  velocities  of  a  particle  of 

water  at  M,  just  before  striking  the  vane; 
Y2j  V2  =  absolute  and  relative  velocities  of  a  particle  at 

N,  just  leaving  the  vane; 
AI,  a\  =  angles  between  FI,  v\  respectively  and  u\ , 

A  <*  ll  T7ii  ((  (f        ni 

Az,  a2  =  v  2,  V2  u%. 

In  order  to  compute  the  action  of  the  jet  upon  the  vane, 
we  may  apply  the  principle  of  angular  momentum,  taking  the 
axis  of  rotation  0  as  axis  of  moments.  Let  OM  =  ri,  ON  =  r2. 

The  angular  momentum  of  a  particle  of  water  of  mass  m 
just  before  striking  the  vane  is 

mV\r\  cos  A\, 

*By  "  relative  velocity"  will  always  be  meant  "  velocity  relative  to  the 
rotating  body,"  as  described  in  Appendix  B. 


FORCE  EXERTED   BY  CONFINED  STREAM.  169 

and  its  angular  momentum  just  as  it  leaves  the  vane  is 

mV2r2  cos  A2, 

so  that  the  action  of  the  vane  changes  the  angular  momentum 
of  each  particle  by  the  amount 

m(V2r2  cos  A2  —  V\r\  cos  A\) . 

The  total  change  of  angular  momentum  due  to  the  action  of 
the  vane  for  one  second  is 


— (V2r2  cos  A2  —  V\r\  cos  AI), 

if  Wf  is  the  weight  of  water  striking  the  vane  per  second.  This 
is  therefore  also  the  average  value  of  the  total  moment  of  the 
forces  exerted  by  the  vane  upon  the  water. 

If  there  is  a  succession  of  similar  vanes  forming  a  water 
wheel,  the  total  action  upon  the  wheel  is  found  by  replacing 
Wr  by  W,  the  total  weight  of  water  used  by  the  wheel  per 
second.  If  G  denotes  the  total  moment  of  the  forces  exerted 
by  the  water  upon  the  wheel,  we  have 

W 
—G= — (V2r2  cos  A2  —  ViTi  cos  AI), 

y 

W 

or  G=— (Fin  cos  AI—  V2r2  cos  A2).     .   ..    .     (19) 

Applications  of  this  result  will  be  given  in  the  following  chapters. 

170.  Force  Exerted  upon  Pipe  by  Confined  Stream  in  Steady 
Flow. — Let  A  B  (Fig.  87)  represent  a  portion  of  a  pipe  carrying 
a  steady  stream,  FI,  FI  being  the 
cross-section  and  velocity  at  A  and 
F2,  V2  the  like  quantities  at  B.  Let 
the  rate  of  discharge  be  W  pounds  per 
second. 

In  passing  from  A  to  B  a  particle 
of  mass  m  receives  an  increment  of 
momentum   mJF,  in  which   AV  =  A'Bf  (Fig.  87),  FT    and   V-z 
being  represented  by  the  vectors  OAf,  OB'.    Hence  the  average 


170  DYNAMIC  ACTION  OF  STREAMS. 

force  exerted  upon  the  particle  is 

mAV 

~7T' 

if  At  is  the  time  required  for  this  change  of  momentum.  (Art. 
158.)  The  resultant  of  such  average  forces  for  all  particles 
will  be  constant,  and  may  be  computed  by  considering  the  body 
of  water  which  at  any  instant  occupies  the  volume  AB}  and 
following  its  motion  for  the  time  dt.  During  this  time  equal 
weights  of  water  Wdt  pass  the  sections  A  and  B,  the  momentum 

W  W 

of  the  former  being  — Vidt,  and  that  of  the  latter  — V>2dt.    The 

flow  being  steady,  the  vector  difference  between  these  values, 

WAV 

or  -  — dt,  is  the  total  change  of  momentum  of  the  body  under 

consideration  during  the  time  dt.  The  resultant  force  acting 
upon  the  body  is  therefore 

WAV     W 

P=H-2L.=—  (vector  A'ff) (20) 

.7          y 

The  forces  which  make  up  this  resultant  are  the  weight  of 
the  water,  the  pressures  exerted  by  the  adjacent  water  upon  the 
cross-sections  A  and  B,  and  the  forces  exerted  by  the  portion 
of  the  pipe  between  A  and  B.  The  resultant  of  these  latter 
forces  can  therefore  be  determined  if  the  pressures  in  the  stream 
at  A  and  B  are  known.  The  equal  and  opposite  reaction  to 
this  force  is  the  force  exerted  by  the  stream  upon  the  portion 

AB-oi  the  pipe. 

EXAMPLES. 

1.  Let  the  axis  of  the  pipe  in  Fig.  88  be  horizontal,  the  diameter  at 

A.  A  being  1  ft.  and  at  B  .5  ft.     Assume  the  rate 

EJrEfE^Eg^ggg^^^JjjLs  of  discharge  to  be  2  cu.  ft.  per  second  in  the 

^^^^^^^=  direction   AB,   and   the    pressure  head   at   the 

center  of  the  section  A  to  be  12  ft.     Compute 

FIG.  88.  tne  resultant  force  exerted  upon  the  pipe  by  this 

portion  of  the  stream,  on  the  assumption  of  no  loss  of  head. 

From  the  given  data  V^  =2.55,  Fa  =  10.2,  Ft2/20  =  .101,  VS/2g  =  1.62. 
Assuming  no  loss  of  head, 

tL+!£j*+ZL 

w      2g      w      2g' 
from  which  p2/w  — 10.5  ft. 


THEORY  OF  PITOT'S  TUBE.  171 

The  increment  of  momentum  per  second  for  the  body  AB  is 

W(V      V}     62.5  X2X  (10.20  -2.55) 
g(V*     Yl)  ~~32.2 

hence  the  resultant  of  all  forces  acting  upon  this  body  is  29.7  Ibs.  in  the 
direction  AB.  This  is  made  up  of  the  pressures  on  the  two  cross-sections 
A  and  B  and  the  forces  exerted  by  the  pipe.  Calling  the  resultant  of 
the  latter  —  P  (so  that  P  is  the  force  exerted  by  the  stream  upon  the 
pipe),  we  have 


or  P  -piFi  -p*F2  -  29.7  =589  -  129  -  29.7  =430  Ibs. 

2.  Generalizing  the  preceding  example,  show  that  the  resultant  force 
exerted  by  the  stream  upon  the  pipe  A  B,  neglecting  loss  of  head,  is 


-w(Fi 


-'•>[$-(£-)!'•] 


3.  Solve  with  data  as  in  Ex.  1,  except  that  the  flow  is  in  the  opposite 
direction. 

4.  A  stream  is  discharged  into  the  atmosphere   through  a  nozzle 
which  reduces  the  diameter  from  2"  to  .5".     If  the  pressure  head  at 
the  nozzle  entrance  is  100'  above  atmospheric,  compute  the  resultant 
force  exerted  by  the  stream  upon  the  nozzle,  assuming  no  loss  of  head. 

Ans.  1221'bs. 

5.  A  straight  horizontal  pipe  1'  in  diameter  is  discharging  1  cu.  ft. 
per  sec.     Assuming  that  the  loss  of  head  is  given  by  Darcy's  formula, 
determine  the  resultant  force  exerted  by  the  stream  upon  the  pipe  in  a 
length  of  100'.  Ans.  26.7  Ibs. 

171.  Theory  of  Pitot's  Tube.— The  measurement  of  the 
velocity  at  any  point  in  a  stream  by  means  of  Pitot's  tube 
has  been  referred  to  in  Chapter  XIII. 
Fig.  89  represents  such  a  tube  placed 
with  the  lower  end  facing  the  current, 
the  main  part  of  the  tube  being  ver- 
tical and  the  upper  end  projecting  above 
the  water  surface.  Let  V  denote  the 
velocity  of  the  current  passing  the  lower 
end  of  the  tube,  and  h  the  height  to 
which  the  water  in  the  tube  rises  above  FlG-  89- 

the  surface  of  the  stream.    Then  h  measures  the  excess  of  the 


172  DYNAMIC  ACTION  OF  STREAMS. 

pressure  at  any  point  within  the  tube  above  the  pressure  at 
the  same  level  in  the  stream  outside  the  tube.  If  F  is  the 
cross-sectional  area  of  the  tube  at  the  end  facing  the  stream, 
the  dynamic  action  of  the  stream  upon  this  area  must  amount 
to  a  force 

whF 

in  order  to  maintain  the  column  within  the  tube  in  equilibrium. 
The  effect  of  the  tube  is  to  deflect  a  certain  part  of  the  current, 
and  the  total  change  of  momentum  thus  produced  in  one  second 
is  equal  in  magnitude  to  the  force  exerted  upon  the  tube  and 
its  contained  water  by  the  water  which  is  deflected.  (Art.  158.) 
If  W  is  the  weight  of  water  deflected  per  second  and  AV  its 
average  increment  of  velocity,  the  force  would  be 


It  seems  reasonable  to  assume  that  AV  is  proportional  to  F, 
and  that  the  part  of  W  whose  deflection  is  due  to  impingement 
against  the  water  at  rest  in  the  tube  is  proportional  to  wFV, 
so  that  the  force  exerted  by  the  current  upon  the  area  F  is 
proportional  to 

wFV* 
9 

Comparing  with  the  value  given  above, 

wFV* 
whF  varies  as , 

F2 
or  fc-fer, 


in  which  &  is  a  coefficient  which  would  be  constant  if  the  theory 
were  exactly  true. 

The  values  of  k  will  depend  upon  the  form  of  the  tube,  and 


RAM   PRESSURE. 


173 


FIG.  90. 


must  be  determined  by  experiment  for  every  instrument.  For 
the  three  cases  shown  in  Fig.  90  it  would  seem  that  (A) 
would  give  the  least  and  (C)  the 
greatest  value  of  AV  for  a  given 
value  of  V,  and  therefore  that  the 
coefficient  k  would  be  least  in  the 
former  case  and  greatest  in  the 
latter.  This  is  verified  by  experi- 
ment. Values  of  k  ranging  from  1  to  2  have  been  found  by 
different  experimenters  for  tubes  of  different  designs. 

172.  Ram  Pressure. — The  sudden  stoppage  of  the  flow  in 
a  pipe  may  cause  pressures  much  greater  than  those  due  to  the 
static  head.  Pressure  due  to  this  cause  is  called  ram  pressure, 
or  "  water-ram." 

In  Fig.  91  let  the  flow  in  the  pipe  be  steady,  the  velocity 

being  F,  and  suppose  a  valve  at 
C  to  be  suddenly  closed,  thus 
checking  and  finally  stopping 
the  flow.  At  any  point  up- 
stream from  the  valve  the  pres- 
sure is  increased  by  an  amount  which  is  less  as  the  distance 
from  the  valve  is  greater.  The  amount  of  increase  at  any 
point  depends  upon  the  rate  of  decrease  of  the  velocity.  This 
effect  is  strictly  analogous  to  that  due  to  a  moving  body 
which  is  brought  to  rest  by  striking  a  fixed  obstacle;  it  exerts 
upon  the  obstacle  a  force  proportional  directly  to  the  mass  of 
the  moving  body  and  to  the  rate  at  which  its  velocity  is  de- 
creased. The  value  of  the  ram  pressure  at  any  point,  in  the 
ideal  case  of  an  incompressible  fluid  and  an  unyielding  pipe, 
may  be  expressed  in  the  following  manner. 

Let  AB  be  any  straight  portion  of  the  pipe  of  length  Z,  and 
let  pi,  p2  denote  the  values  of  the  pressure  at  A  and  B.  If  m 
is  the  total  mass  of  the  cylinder  of  water  AB,  the  resultant  of 
all  forces  acting  upon  it  at  any  instant  is  (by  the  fundamental 
equation  of  motion) 


FIG.  91, 


(22) 


174  DYNAMIC  ACTION  OF  STREAMS. 

While  the  flow  is  steady  this"  resultant  is  zero,  since  dV/dt=Q, 
but  while  the  velocity  is  changing  P  is  not  zero.  In  the  present 
problem  the  forces  called  into  action  by  the  sudden  closing  of 
the  valve  are  great  in  comparison  with  those  acting  when  the 
flow  is  steady,  and  the  latter  may  be  neglected  as  in  all  cases 
in  which  impulsive  forces  are  considered.*  The  only  forces  to 
consider  are,  therefore,  the  pressures  caused  by  the  sudden 
stoppage  of  the  flow.  The  resultant  of  these  pressures  acting 
upon  the  body  AB  amounts  to  a  force 


in  the  direction  AB,  F  being  the  cross-section  of  the  pipe.     But 
also,  as  above, 

dV    wFl  dV 


wl  dV 
therefore  p2  -  pl  =  ---  — 

i/ 


.-L=--          .......     (24) 

w     w         g    dt 

Since  dV  /dt  is  negative  (because  V  decreases)  ,  the  second  mem- 
ber of  this  equation  is  really  positive,  and  p2  is  greater  than  p\  . 

This  formula  shows  that  the  ram  pressure  varies  along  the 
pipe  directly  as  the  length,  and  is  independent  of  the  diameter 
for  a  given  value  of  dV  /dt.  The  result  is  seen  to  hold  even  if 
the  pipe  is  curved,  since  the  length  may  be  subdivided  into 
elements  in  applying  the  above  reasoning. 

If  the  pipe  leads  from  a  reservoir,  as  in  Fig.  91,  the  ram 
pressure  will  be  zero  at  the  intake  end,  so  that  for  a  point  dis- 
tant I  from  the  intake  end  we  may  write  (pressure  being  ex- 
pressed in  terms  of  equivalent  water  column) 

I   fly 
Ram  pressure  =  ---  -77  ......     (25) 

*  Theoretical  Mechanics,  Art.  321  ! 


RAM  PRESSURE.  175 

No  practical  use  can  be  made  of  this  formula  unless  dV/dt 
can  be  estimated.  Experiment  indicates  that  the  closing  of  a 
valve  produces  no  ram  pressure  of  importance  except  during 
the  last  part  of  the  closing;  that  is,  the  rate  of  change  of  the 
velocity  does  not  become  great  until  near  the  very  end  of  the 
process.  Dangerous  ram  pressures  may  therefore  be  avoided 
by  using  valves  so  constructed  that  the  last  part  of  the  closing 
must  be  slow. 

The  above  results  are  doubtless  modified  in  an  important 
degree  by  the  elastic  yielding  of  the  pipe  and  of  the  water.  It 
is  doubtful  whether  a  theoretical  discussion  taking  account  of 
these  factors  can  be  given  which  will  accord  closely  with  prac- 
tical conditions. 

EXAMPLE. 

1.  If  the  velocity  of  flow  in  a  pipe  is  changed  in  .25  sec.  from  2  ft. 
per  sec.  to  0  by  the  closing  of  a  valve  1000  ft.  from  the  reservoir,  esti- 
mate the  average  value  of  the  ram  pressure  close  to  the  valve. 

Ans.  p/w=248  ft. 


CHAPTER  XV. 
THEORY  OF  STEADY  FLOW  THROUGH  ROTATING  WHEEL. 

173.  Statement  of  Problem.  —  Wheels  designed  for  the 
utilization  of  the  energy  of  streams  of  water  are  of  various 
forms,  and  may  be  divided  into  several  classes  possessing  some- 


FIG.  92. 

what  distinct  characteristics.  The  same  basal  theory,  how- 
ever, applies  to  practically  all  the  important  types.  This 
theory  it  is  the  object  of  the  present  chapter  to  present;  and  for 
illustration  reference  will  be  made  to  the  arrangement  shown  in 
Fig.  92. 

The  figure  shows  an  elevation  and  vertical  section  (A)  and 

176 


NOTATION.  177 

a  plan  and  horizontal  section  (B).  The  "  penstock  "  or  supply- 
pipe  P  leads  from  the  head  race  or  supply  reservoir  and  ter- 
minates in  the  guide  passages  g  g.  These  are  so  formed  as 
gradually  to  deflect  the  water  outward  and  forward  (i.e.,  in  the 
direction  of  the  rotation).  A  particle  of  water  when  leaving 
a  guide  passage  is  moving  in  a  horizontal  plane  and  in  a  direc- 
tion determined  by  that  of  a  guide  vane.  Within  the  wheel 
the  path  of  a  particle  relative  to  the  wheel  is  determined  by  the 
form  of  the  wheel  vanes.  Arriving  at  the  outlet  of  the  wheel, 
the  relative  velocity  of  a  particle  has  a  direction  determined  by 
that  of  a  wheel  vane,  while  its  absolute  velocity  depends  upon 
this  relative  velocity  and  also  upon  the  rotation  of  the  wheel. 

The  problem  whose  solution  is  the  basis  of  turbine  theory 
is  the  following:  Given  all  dimensions  and  the  rotation  speed 
of  the  wheel,  required  to  determine  the  rate  of  flow  through  the 
wheel. 

If  the  wheel  were  at  rest,  this  would  be  a  problem  in  ordinary 
hydraulics,  to  be  solved  by  the  methods  illustrated  in  Chapter 
VI.  It  is  to  be  shown  how  the  rotation  of  the  wheel  affects 
the  solution.  The  explanation  involves  the  following  points : 

(1)  The  relation  between  the  absolute  and  relative  velocities 
of  a  particle;  (2)  the  meaning  of  the  general  equation  of  energy 
in  such  a  case  as  this;  (3)  the  computation  of  the  energy  given 
up  by  the  stream  to  the  wheel;  (4)  the  deduction  from  these 
results  of  a  special  form  of  the  energy-equation  involving  rela- 
tive instead  of  absolute  velocities. 

In  the  present  chapter  this  problem  will  not  be  completely 
solved,  since  the  details  of  the  solution  must  be  different  for 
different  types  of  wheel.  In  all  cases,  however,  the  solution 
involves  certain  general  steps  which  are  here  outlined. 

174.  Notation.— The  following  notation  will  be   employed 
throughout  the  entire  discussion  of  turbines  and  water  wheels : 
r— distance  of  a  particle  from  the  axis  of  rotation; 
2= its  height  above  a  horizontal  datum  plane; 
v=its  velocity  relative  to  the  wheel; 
F=its  velocity  relative  to  the  earth; 


178          STEADY   FLOW  THROUGH   ROTATING  WHEEL. 

[Briefly,  v  and  V  will  usually  be  called  simply  relative  velocity 
and  absolute  velocity.] 

aj  =  angular  velocity  of  wheel  (radians  per  second) ; 
u  =  raj  =  linear    velocity    of    point    of    wheel    momentarily 
coinciding  with  a  particle  of  water  under  considera- 
tion; 
s  =  resolved  component  of  V  in  direction  of  u  (called  briefly 

tangential  component  of  V) ; 
a  =  angle  between  v  and  u\ 
A  =  angle  between  V  and  u\ 
q  =  volume  of  water  discharged  per  second ; 
W  =  weight  of  water  discharged  per  second; 
w= weight  of  unit  volume  of  water; 
F  =  cross-section  of  stream  at  any  point  in  the  stationary 

part  of  the  passages; 

/  =  cross-section  of  stream  at  any  point  within  the  wheel; 
L  =  energy  imparted  by  water  to  rotating  wheel  per  second. 
The  values  of  any  of  the  foregoing  quantities  for  different 
sections  of  the  stream  will  be  distinguished  by  suffixes.    Thus 
suffix  d)  will  refer  to  the  stream  leaving  the  guide  passages, 
just  before  its  motion  is  influenced  by  the  wheel,  and  suffix  (2) 
to  the  stream  leaving  the  wheel. 

In  many  cases  the  stream  is  divided  into  several  parts  in 
passing  through  the  wheel,  also  as  it  approaches  the  wheel 
through  the  guide  passages.  By  F  and  /  will  be  meant  the 
sum  of  the  cross-sections  of  all  the  partial  streams  taken  at 
corresponding  points  in  the  different  passages.  Thus 

FI  =  total  cross-section   of  streams  leaving  guide  passages, 

measured  everywhere  normally  to  the  direction  of  FI  ; 

/2  =  total  cross-section  of  streams  leaving  wheel  passages, 

measured  everywhere  normally  to  the  direction  of  v2. 

The  equation  of  continuity  (Art.  36)  holds  throughout  the 

entire  series  of  passages;  thus 


APPLICATION  OF  GENERAL  EQUATION   OF  ENERGY.     179 

In  considering  the  relation  between  absolute  and  relative 
velocities  it  is  often  convenient  to  use  the  language  of  vector 
addition.  For  this  purpose  the  vector  value  of  a  velocity  will 
be  represented  by  the  use  of  brackets:  [u],  [v],  [V]  =  vector 
values  of  velocities  whose  magnitudes  are  u,  v,  V. 

175.  Relation  between  Absolute  and  Relative  Velocities.*  — 

The  three  velocities  u,  v,  V  are  related  in  the  manner  shown 
by  the  vector  triangle  in  Fig.  93.    This  relation 
is  expressed  by  the  statement  that  V  is  the  vector 
sum  of  u  and  r,  or 


......     (1) 

Algebraically,  any  of  the  equations  given  by 
elementary  trigonometry  may  be  used   for   com- 
puting one  of  these  vectors  when  the  other   two   are   known. 
The  following  are  especially  useful.    Resolving  along  and  per- 
pendicularly to  u  (Fig.  93), 

V  cos  A  =  u  +  v  cos  a,  1 
V  sin  A  =  v  sin  a.         j  ' 

The  tangential  component  of  V  is  evidently 

s  =  V  cos  A  =  u  +  v  cos  a  .......     (3) 


These   relations    being   true    for  any  particle,  are   true  when 
suffix  d)  or  suffix  (2)  is  attached  to  each  symbol. 

176.  Application   of    General   Equation   of  Energy.  —  The 

general  equation  of  energy 


may  be  applied  to  any  case  of  steady  flow.  In  this  equation 
H'  means  the  quantity  of  energy  lost  by  the  water,  between 
the  sections  d)  and  (2),  per  pound  of  water  discharged.  (Art. 
63.)  In  most  of  the  applications  hitherto  considered  this  loss 
has  been  wholly  due  to  dissipation  of  energy  into  heat:  If  the 
water  gives  up  energy  in  any  other  way,  the  equation  still  holds, 

*  See  Appendix  B. 


180  STEADY  FLOW  THROUGH  ROTATING  WHEEL. 

provided  H'  be  made  to  include  such  loss  as  well  as  that  due  to 
dissipation  into  heat. 

Thus,  in  the  case  shown  in  Fig.  92,  let  the  section  d)  be 
taken  where  the  water  is  leaving  the  guide  passages,  just  before 
its  motion  is  influenced  by  the  rotating  wheel,  and  the  section 
(2)  at  the  point  of  outflow  from  the  wheel.  Then  H'  is  made 
up  of  two  parts,  corresponding  respectively  to  the  energy  lost 
by  dissipation  within  the  wheel  and  that  given  up  to  the  wheel 
as  mechanical  energy.  Calling  these  parts  hr  and  h"  respect- 
ively, we  have 

1  =  2g  +¥    Zl> 


and  the  equation  may  be  written 

+*  -+•  -*+*••  •  •  «> 


The  equation  applies  also  if  the  section  d)  be  taken  elsewhere, 
for  example  at  the  surface  of  the  head  race,  proper  values  being 
given  to  FI,  pi  and  z\,  and  it  being  understood  that  h'  includes 
energy  dissipated  throughout  the  entire  series  of  passages 
between  d)  and  (2),  while  the  meaning  of  h"  is  unchanged.* 

177.  Computation  of  Energy  Given  Up  to  Wheel.  —  The 
value  of  the  energy  given  up  to  the  wheel  by  the  water  may 
be  expressed  by  a  simple  formula  which  will  now  be  deduced. 

The  flow  being  steady  and  the  rotation  of  the  wheel  uniform, 
the  water  exerts  upon  the  wheel  forces  whose  turning  moment 
about  the  axis  of  rotation  is  constant.  If  G  denotes  this  mo- 
ment, the  work  done  by  the  forces  upon  the  wheel  f  while  the 
latter  turns  through  an  angle  6  (radians)  is  GO. 

*  See  Art.  94.  f  Theoretical  Mechanics,  Art.  508. 


NEW  FORM  OF  GENERAL  EQUATION  OF  ENERGY.   181 

The  value  of  G  may  be  determined  from  the  principle  of 
angular  momentum,  as  in  Art.  169.  The  value  there  found 
may  be  written  (noticing  that  FI  cos  AI  =Si  and  ¥2  cos  A2  =  s2) 

W 

G  =—(r1s1-r2s2)  .......     (5) 

t/ 

The  angle  turned  through  in  one  second  being  to,  the  work 
done  on  the  wheel  per  second  is  Ga>;  or,  denoting  by  L  the 
energy  imparted  to  the  wheel  in  one  second,  and  noticing  that 


W  W 

L=  —  (riSi—r2s2)to  =  —  (uis\—  u2s2).     ...     (6) 

y  y 

178.  New  Form  of  General  Equation  of  Energy.  —  From  the 
meaning  of  h"  (Art.  176)  it  is  evident  that 


so  that 


and  equation  (4)  may  be  written 


This  may  be  simplified  by  introducing  relative  instead  of  abso- 
lute velocities.  Thus,  from  the  vector  relation  between  F,  v 
and  u  (Fig.  93), 


V22  =  v22  +  u22  -f  2u2v2  cos  a2j 
SI=MI  +vi  cos  ai, 

COS  C12. 


182  STEADY  FLOW  THROUGH   ROTATING  WHEEL. 

The  substitution  of  these  values  in  (7)  reduces  it  to  the  form 


•     (8) 


This  may  be  called  the  equation  of  energy  for  the  relative 
motion. 

It  may  be  noticed  that  this  equation  includes  as  a  special 
case  the  equation  of  energy  in  its  ordinary  form  for  flow 
through  stationary  passages,  reducing  to  it  when  o;  =  0; 
this  makes  HI  and  u2  zero,  and  v\  and  v2  become  absolute 
velocities. 

The  meaning  of  h'  should  be  kept  clearly  in  mind.  It  is 
the  energy  lost  by  dissipation  within  the  wheel,  per  pound  of 
water  discharged. 

179.  Summary  of  Principles.  —The  principles  to  be  employed 
in  the  theory  of  turbines  are  embodied  in  the  general  formulas 
already  deduced,  which  may  be  summarized  here  for  convenience 
of  reference. 

The  equation  of  continuity: 


q  =  FV  =  fv  =  constant;      .....      (I) 
and  in  particular 

(I') 


The  relation  between  absolute  and  relative  velocities  of  a 
particle,  expressed  by  the  vector  equation 


and  by  the  algebraic  equations 


V  cos  A  =  u  +  v  cos  a,  1 

Ft  '  ( 

sin  A  =  v  sin  a, 


the  two  algebraic  equations  (II')  being  equivalent  to  the  single 
vector  equation  (II) . 


EXAMPLES.  183 

The  equation  of  energy  for  steady  flow  in  the  ordinary  form: 


The  formula  for  energy  imparted  to  the  wheel: 

W 
L  =—(ulsl-u2s2)  .......     (IV) 

The  equation  of  energy  for  the  relative  motion: 


180.  Application  of  General  Theory.  —  The  application  of  the 
foregoing  principles  and  formulas  to  particular  cases  involves 
special  data  pertaining  to  each  case.  The  two  cases  of  greatest 
practical  importance  are  illustrated  in  the  following  examples, 
and  the  complete  theory  is  treated  in  the  succeeding  chapters 
relating  to  the  different  types  of  turbines  and  water  wheels. 

The  following  examples  refer  to  the  arrangement  shown  in 
Fig.  92. 

EXAMPLES. 

1.  A  particle  of  water  leaves  the  guide  passage  with  a  velocity  of  22 
ft.  per  sec.  directed  at  angle  18°  with  the  velocity  u\.     The  distance 
from  the  axis  of  rotation  is  3.5  ft.,  and  the  wheel  rotates  at  the  rate  of 
90  R.P.M.     Determine  the  magnitude  and  direction  of  the  velocity  of 
the  particle  relative  to  the  wheel. 

Am.  v,  =13.85  ft.  per  sec.;  a,  =  150°  36'. 

2.  In  order  that  the  particle  (Ex.  1)  shall  not  be  suddenly  deflected 
as  it  enters  the  wheel,  what  should  be  the  direction  of  the  wheel  vane  at 
the  point  where  the  water  enters? 

3.  Using  the  data  of  Ex.  1,  and  assuming  that  the  particle  moves 
in  a  horizontal  plane,  that  no  frictional  loss  of  energy  occurs  within  the 
wheel,  and  that  the  pressure  is  constant  throughout  the  wheel  and  at 
guide  outlets,  what  will  be  the  magnitude  of  the  relative  velocity  of  a 
particle  when  4  ft.  from  the  axis?     What  will  determine  the  direction  of 
this  relative  velocity?  Ans.  v  =22.91  ft.  per  sec. 


184  STEADY  FLOW  THROUGH  ROTATING  WHEEL. 

4.  Suppose  the  radius  of  the  wheel  at  outlet  is  4.5  ft.,  and  a2  =  162°. 
Making  the  same  assumptions  as  in  Ex.  3,  determine  the  magnitude  and 
direction  of  the  relative  velocity  and  of  the  absolute  velocity  of  a  par- 
ticle leaving  the  wheel. 

Ans.  v2=  30.05  ft.  per  sec.;  V.  =  16.66  ft.  per  sec.;  A2=33°  52'. 

5.  With  data  as  in  Ex.  4,  (a)  how  much  energy  does  the  wheel  receive 
from  the  water  for  each  pound  of  discharge?     (6)  If  F\  =6  sq.  ft.,  com- 
pute the  H.P.  imparted  to  the  wheel,     (c)  The  energy  utilized  is  equiva- 
lent to  what  fall?  Ans.  (a)  3.21  foot-pounds. 

[In  the  following  examples,  instead  of  supposing  the  pressure  to  be 
uniform  throughout  the  wheel,  assume  that  the  wheel  passage  is  com- 
pletely filled  at  every  cross-section,  and  that  F,//2  =  1.5,  all  other  data 
being  as  in  preceding  examples.  Answer  the  following  questions.] 

6.  What  further  data  are  required  for  the  determination  of  the  rela- 
tive and  absolute  velocities  of  a  particle  4  ft.  from  the  axis? 

7.  Determine  the  magnitude  and  direction  of  the  absolute  velocity 
and  of  the  relative  velocity  of  a  particle  as  it  leaves  the  wheel. 

Ans.  ^=33  ft.  per  sec.;  F2  =  15.03  ft.  per  sec.;  A2=42°  43'. 

8.  If  the  discharge  from  the  wheel  takes  place  into  the  atmosphere, 
what  pressure  exists  at  guide  outlets?  Ans.  p\/w  —  p^/w  =2.9  ft. 

9.  If  F>  =6  sq.  ft.,  compute  H.P.  imparted  to  wheel.    What  is  the 
value  of  the  "utilized  head  "? 

Ans.  H.P.  =103.5;  utilized  head  =6.90  ft. 


CHAPTER  XVI. 
TYPES  OF  TURBINES  AND  WATER  WHEELS. 

181.  Definition  of  Turbine.  —  There  is  no  general  agreement 
upon  an  exact  definition  of  a  turbine.  By  Bodmer  *  a  turbine 
is  defined  as  "a  water  wheel  in  which  a  motion  of  the  water 
relatively  to  the  buckets  is  essential  to  its  action."  Such  a 
definition  includes  practically  all  modern  wheels  for  the  utiliza- 
tion of  water  power.  The  same  author  uses  the  term  water 
wheel  to  designate  the  old-fashioned  overshot,  undershot,  and 
breast  wheels,  which  were  usually  of  large  diameter  relatively 
to  the  fall  utilized,  and  which  were  actuated  either  by  the 
weight  of  the  water  directly  or  by  the  impact  of  a  stream  against 
flat  vanes. 

These  definitions  are  not  in  conformity  with  present-day 
usage  in  America.  This  usage  can  be  best  explained  after  the 
different  types  of  wheels  have  been  described.  f  The  word  tur- 
bine will,  however,  generally  be  used  with  the  broad  meaning 
above  given. 


Classification  of  Turbines  According  to  Direction  of 
Flow.  —  The  direction  of  flow  of  the  water  through  a  turbine 
may  be  either  radial,  axial,  or  mixed. 

Radial  flow  means  that  the  path  of  a  particle  within  the 
wheel  lies  in  a  plane  perpendicular  to  the  axis  of  rotation.  The 
direction  of  flow  may  be  either  outward  (Fig.  92)  or  inward 
(Fig.  105). 

Axial  flow  means  that  the  distance  of  a  particle  from  the 

*  Hydraulic  Motors,  p.  24.  t  See  Art.  191. 

185 


186  TYPES  OF  TURBINES  AND   WATER  WHEELS. 

axis  of  rotation  remains  constant  during  its  passage  through 
the  wheel  (Fig.  104). 

Mixed  flow  is  a  combination  of  radial  and  axial  flow.  It 
is  usually  inward  and  axial,  as  in  Fig.  106. 

183.  Classification  into  Impulse  Wheels  and  Reaction  Wheels. 
— If  the  total  cross-section  of  the  streams  entering  the  wheel 
is  so  small  that   the  wheel  passages  are  not  filled,  and  if  air 
enters  freely  so  that  the  entire  stream  within  each  wheel  pas- 
sage is  under  atmospheric  pressure,  the  turbine  is  called  an 
impulse  wheel. 

If  the  wheel  passages  are  completely  filled  by  the  streams 
flowing  through  them,  the  turbine  becomes  a  reaction  wheel. 

This  is  the  most  fundamental  distinction  between  different 
wheels  of  modern  type. 

184.  Complete    and    Partial    Admission. — Water    may    be 
admitted  to  all  the  wheel  passages  at  once,  or  to  a  limited  num- 
ber   of    them.    Reaction    wheels    necessarily    have    complete 
admission,  while  impulse  wheels  may  have  partial  admission 
with  little,  if  any,  sacrifice  of  efficiency. 

185.  Conditions  of  Discharge. — Impulse  wheels  always  dis- 
charge into  the  air.    A  reaction  wheel  may  discharge  either  (a) 
into  the  air,  (6)  into  a  body  of  free  water,  or  (c)  into  a  suction 
tube. 

When  the  discharge  is  into  the  atmosphere,  the  fall  from 
the  point  of  discharge  to  the  tail  race  is  lost.  Such  loss  does 
not  occur  if  the  wheel  is  submerged,  or  if  it  discharges  into  a 
suction  tube  (provided  the  conditions  are  such  that  water  fills 
the  tube  throughout  its  length) . 

The  action  of  a  suction  tube  depends  upon  atmospheric 
pressure.  The  pressure  in  the  tube  at  the  upper  end  (at  Y, 
Fig.  104)  is  less  than  atmospheric  by  an  amount  depending 
upon  the  height  of  this  point  above  the  surface  of  the  tail  race. 
This  decrease  in  the  pressure  at  the  point  where  the  wheel  dis- 
charges has  the  same  effect  upon  the  flow  as  an  equal  increase 
at  X,  the  point  of  inflow.  The  fall  z  (Fig.  104)  below  the  wheel, 


DIFFERENT  FORMS  OF  TURBINES.  187 

which  would  be  lost  if  the  discharge  occurred  at  the  same  point 
but  under  atmospheric  pressure,  is  thus  exactly  compensated. 
If  the  wheel  be  placed  lower  or  higher,  the  pressures  at  X  and 
Y  will  change  equally,  so  that  the  operation  of  the  wheel  will 
be  unchanged.  Of  course  z  must  not  be  so  great  that  the 
pressure  at  Y  is  reduced  to  absolute  zero.* 

186.  Girard    Impulse    Turbine. — Impulse    turbines    of    the 
type  known  as  the  Girard  have  been  extensively  used  in  Europe, 
also  to  some  extent  in  America.    The  flow  may  be  either  radial 
or  axial,  and  admission  may  be  either  complete  or  partial. 

The  case  of  radial  flow  with  complete  admission  may  be 
represented  as  regards  general  arrangement  by  Fig.  95.  The 
arrangement  of  a  Girard  turbine  with  axial  flow  is  shown  in 
Fig.  96. 

Girard  introduced  the  feature  of  ventilating  the  wheel 
passages  by  orifices  for  the  free  admission  of  air. 

187.  American  Tangential  Water  Wheels. — The  type  of  motor 
shown  in  Figs.  98  and  99  is  common  in  America,  especially 
in  mountainous  regions  where  high  falls    are   utilized.      The 
wheel  vanes  or  buckets  receive  the  water  in  a  cylindrical  stream 
from  a  nozzle,  the  axis  of  the  stream  being  tangent  to  the  circle 
described  by  a  certain  point  in  each  bucket.     Buckets  have 
been  made  of  many  forms,  but  in  most  cases  the  "  split  "  bucket 
is  used,  the  stream  being  divided  by  a  sharp  edge  into  two 
streams  which  follow  the  opposite  but  similar  bucket  surfaces. 
Fig.  97  represents  a  section  of  such  a  bucket  by  a  plane  parallel 
to  the  axis  of  rotation  and  to  the  stream  from  the  nozzle. 

Although  differing  greatly  in  form  from  the  turbines  above 
mentioned,  the  tangential  water  wheel  falls  under  the  defini- 
tion of  turbine  above  given,  and  is  of  the  impulse  type  with 
approximately  axial  flow. 

188.  Fourneyron  Turbine. — This  name  is  usually  given  to 
reaction   turbines   with   radial   outward   flow.    The   discharge 
may  be  either  into  the  air  or  into  a  body  of  water.    In  either 

| 

*  A  discussion  of  the  suction  tube  is  given  in  Art.  228. 


188  TYPES  OF  TURBINES  AND  WATER  WHEELS. 

case  the  general  arrangement  may  be  represented  by  Fig.  92. 
A  suction  tube  cannot  conveniently  be  used  with  this  form  of 
wheel. 

189.  Reaction  Turbines  with  Inward  Flow. — Wheels  similar 
in  principle  to  the  Fourneyron  but  with  inward  flow  have  been 
designed  by  Thomson,  Francis,  and  others.     An  inward-flow 
wheel  with  suction  tube  is  shown  in  Fig.  105.    This  type  is  now 
quite  generally  known  as  the  Francis  turbine. 

190.  Jonval     Turbine. — This    is    a   reaction    turbine    with 
axial  flow.    The  discharge  may  be  either  into  the  air,  directly 
into  the  tail  water,  or  into  a  suction  tube  (Fig.  104). 

191.  American    Reaction    Turbines.  —  The   most   common 
forms  of  reaction  turbine  used  in  America  are  of  the  mixed- flow 
type,  having  inward  admission  and  axial  discharge  (Fig.  106). 

In  the  United  States  the  name  turbine  is  usually  confined 
to  the  wheels  above  called  reaction  turbines,  while  the  "  tan- 
gential "  wheels  described  in  Art.  187  are  called  water  wheels. 

192.  Theory    of   Turbines.  —  The   principles   developed    in 
Chapter  XV  furnish  a  basis  for  the  theory  of  turbines  and 
water  wheels  of  all  the  foregoing  types.     In  the  following  chap- 
ters will  be  given  the  outline  of  the  theory  for  the  impulse 
turbine,  the  American  tangential  water  wheel,  and  the  reaction 
turbine.     A  discussion  will  also  be  given  of  turbine  pumps,  of 
which  the  theory  will  be  seen  to  be  closely  similar  to  that  of 
reaction  turbines. 

The  following  definitions  of  available  energy  and  efficiency 
apply  to  all  types  of  water  wheels  and  turbines  acting  as  motors, 
but  not  to  turbine  pumps. 

193.  Available   Energy. — The  available  energy  of  a  stream 
of  water  *  depends  upon  the  weight  of  the  water  and  the  avail- 

*  Strictly  this  energy  is  possessed  not  by  the  water  alone  but  by  the 
system  consisting  of  the  water  and  the  earth,  being  due  to  the  gravitational 
attraction  between  the  earth  and  the  water.  (Theoretical  Mechanics,  Art. 
362.)  The  kinetic  energy  due  to  the  velocity  of  flow  in  the  stream  is  so  small 


EFFICIENCY.  189 

able  fall;  thus  Wh  foot-pounds  of  potential  energy  are  given 
up  by  W  pounds  of  water  in  descending  h  feet.  The  object  of 
a  motor  is  to  receive  this  from  the  water  as  mechanical  energy, 
with  as  little  loss  by  dissipation  as  possible. 

The  available  power  is  the  energy  available  per  unit  time.  If 
the  rate  of  discharge  of  the  stream  is  q  cu.  ft.  per  sec.  and  the 
fall  h  ft.,  the  available  power  is 

wqhft.-lbs.  per  sec.=T      H.P. 


194.  Efficiency.  —  In  estimating  efficiency  we  may  be  con- 
cerned with  the  whole  apparatus,  including  the  pipe  or  channel 
leading  to  the  motor,  the  motor  proper,  and  the  pipe  (if  any) 
leading  from  the  motor  to  the  tail  race,  or  we  may  be  con- 
cerned only  with  the  efficiency  of  the  motor  proper.  Again, 
we  may  consider  the  energy  imparted  to  the  motor  by  the 
water  to  be  the  useful  effect,  or  we  may  regard  the  energy  ob- 
tained from  the  motor  in  doing  external  work  as  the  useful 
effect.  In  any  case,  the  efficiency  is  defined  by  the  equation 

energy  utilized 
e  == 


energy  given  up  by  water' 

but  the  numerator  and  denominator  may  each  have  two  differ- 
ent values,  according  to  the  point  of  view  as  just  explained. 

Gross  efficiency  is  computed  by  taking  the  external  work 
done  by  the  motor  as  "energy  utilized." 

Hydraulic  efficiency^  computed  by  taking  energy  imparted 
to  the  motor  as  "energy  utilized." 

These  two  efficiencies  differ  because  of  the  energy  dissipated 
by  mechanical  friction  of  the  moving  parts  of  the  motor.  Hy- 
draulic efficiency  corresponds  to  the  "indicated"  efficiency  of  a 
steam  engine,  since  the  indicated  work  is  the  work  actually 
done  on  the  piston  by  the  steam  pressure. 

in  comparison  with  the  potential  energy  due  to  the  fall,  that  it  is  neglected 
in  estimating  the  available  energy. 


CHAPTER  XVII. 
THEORY  OF  IMPULSE  TURBINE. 

195.  Given  Dimensions  and  Data.* — In  the  theory  of  tur- 
bines certain  dimensions  and  other  data  may  be  taken  as  known. 
Thus  in  an  impulse  turbine  Ti,.the  velocity  of  outflow  from 
the  guides,  is  fixed  independently  of  the  construction  of  the 
wheel  and  of  its  speed  of  rotation.  Since  the  discharge  from 
the  guides  occurs  under  atmospheric  pressure,  V\  depends  only 
upon  the  fall  to  the  point  of  outflow  and  the  resistance  occurring 
in  the  passages  leading  from  the  head  race  or  reservoir.  The 
angles  AI  and  a2  will  also  be  taken  as  known.  It  may  be  seen 
that  A i  and  180°  —  0,2  should  both  be  small. 

That  a2  should  approach  180°  is  seen  from  the  fact  that  the 
absolute  velocity  of  outflow  from  the  wheel  (¥2)  should  be  as 
small  as  possible,  since  the  kinetic  energy  represented  by  this 
velocity  is  wholly  lost.  Since  V2  is  the  vector  sum  of  u2  and 
v2,  the  more  nearly  opposite  the  directions  of  u2  and  v2  are  taken 
the  smaller  V2  may  become.  This  require- 
ment is,  however,  limited  by  the  fact  that 
the  smaller  the  angle  180° -a2,  the  smaller 
the  cross-section  of  the  wheel  passages.  Thus, 
let  Fig.  94  represent  a  section  of  several 
vanes  of  an  outward-flow  turbine,  and  let  d2 
I  \/  \  denote  the  vertical  dimension  of  the  wheel 

'  *    \         passages  at  the  point  of  outflow.    If  n  is  the 

total  number  of  vanes,  2nr-2/n  is  the  length 
of   the   portion  of   the  wheel  circumference 
corresponding  to  one  passage,  and  the  cross- 
section  of  a  passage  is  approximately  (2xr2d2  sin  a2)  /n,  so  that 

*  For  notation,  see  Art.  174. 

190 


GIRARD  TURBINE  WITH  RADIAL   FLOW.  191 

approximately 


That  AI  should  be  small  is  seen  from  formula  (IV)  (Art. 
179),  which  gives  the  energy  imparted  to  the  wheel  per  unit 
time.  Neglecting  the  term  u2s2,  which  is  to  be  made  small, 
and  remembering  that  sL  =  Vi  cos  AI,  it  is  seen  that  for  a  given 
power  HI  must  vary  inversely  as  V\  cos  A  i  .  Hence,  to  avoid 
an  excessively  high  wheel  speed,  V\  cos^i  should  be  as  large 
as  practicable.  And  since  V\  is  -fixed  independently  of  the 
design  of  the  wheel,  cos  AI  should  be  as  great  as  practicable. 
This  requirement  is,  however,  limited  by  the  fact  that  F\  de- 
creases as  A\  decreases,  other  dimensions  remaining  the  same. 

In  the  following  theory  the  quantities  taken  as  known  are 
Vi,  AI,  a2,  r2/ri  (which  will  be  called  c),  z\  -z2.  The  discussion 
eads  to  a  formula  for  efficiency,  and  to  the  determination  of 
HI  and  a\  for  highest  efficiency.  These  results  are  found  to  be 
independent  of  the  remaining  dimensions  of  the  wheel. 

(A)  GIRARD  TURBINE  WITH  RADIAL  FLOW. 

196.  General  Arrangement   of  Radial-flow  Girard  Turbine. 

—  The  general  arrangement  of  a  radial-flow  Girard  turbine  with 

axis  vertical  and  with  complete  ad- 

mission  is    represented   in   Fig.   95. 

The  wheel  is  placed  as  near  the  sur- 

face of  the  tail  race  as  practicable. 

While  the  vertical  widening  of  the 

wheel  passages  permits  the  particles 

of  water  to  fall  slightly  during  their 

passage  through  the  wheel,  the  amount 

of  this  fall  will  be  so  small  as  to  be  ==/^^^^^7^^~7^//. 

unimportant,  and  will  be  neglected     /^///////////^/////^/// 

in  the  following  theory.    The  figure 

shows  a  cylindrical  gate  so  placed  as  to  regulate  the  size  of  the 

guide  outlets.    The  value  of  Vi  is  not  materially  changed  by 

varying  the  position  of  the  gate,  so  that  no  serious  loss  of  energy 

results  from  this  method  of  regulating  the  supply  of  water. 


192  THEORY   OF  IMPULSE  TURBINE. 

Girard  turbines  are  often  arranged  for  partial  admission. 
The  various  forms  which  have  been  made,  and  the  devices  for 
regulating  the  admission  of  water  to  the  buckets,  will  not  be 
considered  here,  since  these  modifications  do  not  affect  the 
theory  as  here  presented. 

Governing,  or  the  maintaining  of  a  nearly  uniform  wheel 
speed  in  spite  of  fluctuations  in  the  power  consumed  (or  "load  "), 
is  accomplished  by  means  of  the  regulating  gate.  This  is  con- 
nected by  suitable  mechanism  with  some  form  of  centrifugal 
governor,  so  that  an  increase  in  the  wheel  speed  causes  a  decrease 
in  the  gate  opening. 

197.  Relation  between  Power  and  Wheel  Speed.—  Since  the 
quantity  of  water  used  par  second  is  independent  of  the  speed 
of  rotation,  the  efficiency  is  proportional  directly  to  L,  the 
energy  imparted  to  the  wheel  by  the  water  per  second.  It  is 
necessary,  therefore,  to  express  L  as  a  function  of  tha  velocity 
of  rotation,  and  then  consider  what  value  of  this  velocity  makes 
L  a  maximum.  In  the  general  formula  (IV)  (Art.  179)  the 
variable  velocities  in  the  second  member  may  all  be  expressed 
in  terms  of  u\  in  the  following  manner. 

We  may  write  at  once 


To  determine  s2  it  is  necessary  to  solve  the  problem  of  flow 
through  the  wheel,  which  may  be  done  as  follows: 

From  the  vector  relation  between  V  \,  u\  and  Vi  (Fig.  93), 

vi2  =  V?  +  Ui2  -  2ViUi  cos  Ai  .....     (1) 

The  value  of  v\  thus  determined  is  to  be  substituted  in  formula 
(V)  *  for  the  purpose  of  determining  v2.  Since  in  the  present 
case  z\  =Z2  and  p\  —p2,  formula  (V)  takes  the  form 


» 


It  is  necessary  to  estimate  the  value  of  h'. 

*  Art.  179. 


GIRARD   TURBINE  WITH    RADIAL  FLOW.  193 

The  losses  of  energy  by  dissipation  between  the  sections 
d)  and  (2)  are  due  mainly  to  two  causes:  (a)  the  interference 
of  the  inner  edges  of  the  wheel  vanes  with  the  stream  from 
the  guides,  and  (b)  hydraulic  friction  within  the  wheel.  To 
reduce  the  former  as  much  as  possible  the  edges  of  the  vanes 
against  which  the  stream  impinges  should  be  sharp,  and  the 
direction  of  these  wheel  vanes  should  be  such  as  to  agree  with 
that  of  the  relative  velocity  vi.  This  direction  of  the  wheel 
vanes  can  be  adjusted  to  only  one  particular  value  of  u\,  which 
should  be  that  giving  best  efficiency,  as  yet  unknown.  For 
the  purpose  of  this  discussion,  however,  the  direction  of  the 
vane  may  be  regarded  as  varying  with  u\  so  as  always  to  be 
parallel  to  v\,  leaving  the  actual  value  of  the  vane  angle  to  be 
assigned  when  the  best  value  of  HI  becomes  known.  Loss  (a) 
will  thus  depend  upon  the  relative  velocity  of  the  stream 
entering  the  wheel,  while  loss  (6)  will  depend  upon  the  relative 
velocity  of  flow  through  the  wheel  passages.  At  best  only  an 
approximate  estimate  of  these  losses  is  possible,  and  it  will 
suffice  to  express  the  entire  loss  la'  by  a  single  term: 


(3) 


in  which  &  is  a  coefficient  treated  as  constant,  whose  value 
cannot  be  accurately  estimated  apart  from  experiment.  Equa- 
tion (2)  thus  becomes 


,      .V   .     .     (4) 
which  with  (1)  gives 

.     .     .     .     (5) 


Using  this  value  of  v2,  s2  may  be  expressed  in  terms  of  u\ 
and  constants  as  follows  : 


\r    -L    i    K 


(6) 


194  THEORY  OF  IMPULSE  TURBINE. 

We  now  have  the  values  of  Si,  $2  and  u2,  all  in  terms  of 
MI,  for  substitution  in  formula  (IV).    The  result  is 

W 

L  =  — UiVi  cos  AI  -c2ui2 


c  cos  a2 


Vl 


u1\/V12-2ulVi  cosAi+c2^2.       (7) 


198.  Condition  for  Maximum  Power  and   Efficiency.—  The 
mathematical  condition  for  maximum  L  is  dL/dui=0.    This 
leads  to  an  equation  of  the  fourth  degree  for  determining  u\. 
The  deduction  of  this  equation  and  its  solution  in  particular 
cases  would  involve  a  large  amount  of  labor.     In  view  of  the 
imperfections  in  the  foregoing  theory,  the  following  approxi- 
mate treatment  of  the  problem  is  probably  as  satisfactory  a 
guide  to  design  as  the  exact  solution  would  be. 

The  greatest  value  of  L  results  when  the  losses  of  energy 
are  least.  The  loss  which  varies  most  with  varying  velocity 
is  that  due  to  the  kinetic  energy  of  the  water  leaving  the  wheel. 
Since  V2  is  the  vector  sum  of  u2  and  v2,  it  appears  that  when 
V2  is  small  u2  and  v2  will  be  nearly  equal  in  magnitude.  An 
approximate  solution  of  the  problem  of  maximum  efficiency 
will  therefore  result  from  the  assumption 

V2  —  U2  =  CUi.        -..'.....       (8) 

This  value  of  v2  substituted  in  (5)  gives 

7,2-27^!   COS  At  -&C2Wl2  =  0,      ....       (9) 

which  determines  the  value  of  u\. 

199.  Value  of  Efficiency.  —  The  hydraulic  efficiency  of  the 
wheel  is  the  ratio  of  L  to  the  available  energy  per  unit  time, 
WVi2/2g.    The  efficiency  for  any  velocity  may  thus  be  com- 
puted from  the  general  value  of  L  given  by  (7).    By  substitut- 


GIRARD   TURBINE  WITH   RADICAL  FLOW.  195 

ing  the  value  of  HI  given  by  (9) ,  the  highest  efficiency  (according 
to  the  foregoing  approximate  solution)  is  found.  The  following 
convenient  forms  for  the  values  of  maximum  power  and  effi- 
ciency are  easily  obtained  by  combining  (7)  and  (9) : 

W 

Lm=—[ulViGosAl-c2(l+cosa2)ui2l      .     .    (10) 

t7 

_  O  — 1 

em=2j  Y  cos  4i -c2(l+ 00302)^2 J  >       .     .     (11) 

in  which  Ui/V}  must  have  the  value  given  by  (9). 

200.  Best  Vane  Angle. — The  direction  of  a  wheel  vane  at 
point  of  inflow  should  agree  with  that  of  the  relative  velocity 
i'i  possessed  by  the  water  leaving  the  guides,  in  order  to  prevent 
sudden  deflection  of  the  water  and  consequent  loss  of  energy 
and  decrease  of  efficiency.  Since  the  angle  A\  is  fixed,  the 
vector  triangle  for  u\,  v\,  V\  has  all  its  angles  determined  when 
is  known.  In  fact  equations  (II')  *  give 


u\ 
cotanai  =cotan  A\  —y-  cosec  AI.     .    .    .     (12) 


Simple  approximate  rule.  —  If  k  be  assumed  zero,  equation 
(9)  gives 

Vi  =2ui  cos  AI, 

showing  that  the  triangle  whose  sides  are  Ui,Vi,  V\  is  isosceles, 
Vi  being  equal  to  u\,  and  therefore 


(12a) 

This  is  often  given  as  the  rule  for  best  value  of  a\. 

*  Art.  179. 


196 


THEORY  OF  IMPULSE  TURBINE. 


(B)  GIRARD  TURBINE  WITH  AXIAL  FLOW. 

201.  General  Arrangement. — A  Girard  turbine  with  vertical 

axis  and  axial  flow  is  represented 
in  general  arrangement  in  Fig.  96. 
The  wheel  discharges  as  near  the 
surface  of  the  tail- water  as  prac- 
ticable, and  in  order  that  it  may 
act  strictly  as  an  impulse  wheel 
the  buckets  should  be  provided 
with  orifices  for  the  free  admission 
of  air.  The  features  which  dis- 
tinguish this  case  from  the  pre- 
ceding, as  regards  the  theory,  are 
the  equality  of  r\  and  r2,  and  the  inequality  of  z\  and  z2. 

202.  Best  Velocity. — Following  the  same  general  method  as 
in  the  case  of  radial  flow,  formula  (V)  now  takes  the  form 


FIG.  96. 


___ 


(13) 


which  replaces  equation  (2).  Making  the  same  assumption  as 
to  the  value  of  hf  as  in  the  preceding  case,  and  also  assuming 
that  u2  =  v2  for  highest  efficiency,  we  are  led  to  the  equation 

in  which  u  is  written  for  each  of  the  equal  velocities  u\  and  u2. 
This  replaces  equation  (9) . 

In  many  cases  21—22  is  a  small  fraction  of  the  total  fall 
and  may  be  neglected;  but  for  very  low  falls  it  must  be  retained. 

203.  Efficiency. — The  available  energy  in  this  case  includes 
that  due  to  the  fall  within  the  wheel,  2i-22;  therefore  when 
this  is  an  important  fraction  of  the  total  fall,  the  general  for- 
mula for  efficiency  is 

L 

& -IT    n  1 


GIRARD  TURBINE  WITH   AXIAL   FLOW.  197 

and  the  maximum  efficiency  is  found  by  using  the  value  of  L 
corresponding  to  the  best  speed  as  computed  from  (14).  This 
may  be  written  in  the  form 

W 

Lm=—[uVi  cos  AI  -  (1  +cos  a2)u2].    .    .    .    (15) 
y 

Equations  (14)  and  (15)  correspond  to  (9)  and  (10)  of  the  pre- 
ceding case. 

204.  Best  Vane  Angle. — When  z\—z%  is  not  negligible,  the 
best  value  of  the  vane  angle  depends  not  merely  upon  the 
ratio  of  u  and  V\  (as  in  the  preceding  case),  but  upon  their 
absolute  values,  since  equation  (14)  does  not  give  a  value  of 
u/Vi  which  is  independent  of  the  actual  value  of  V\.  The 
best  value  of  the  vane  angle  cannot,  therefore,  be  determined 
except  by  assuming  the  ratio  of  z\—Z2  to  Vi2/2g.  Strictly 
speaking,  then,  a  given  wheel  cannot  work  with  equal  efficiency 
under  different  falls.  This  consideration  is  of  little  importance 
if  the  fall  exceeds  a  very  few  feet. 

When  u  and  V\  are  known,  a\  may  be  computed  from  equa- 
ion  (12). 

EXAMPLES. 

1.  Take  data  as  follows  for  an  impulse  turbine  with  radial  outward 
flow:  Ai  =20°,  a?  =  160°,  r,/r,  =1.2.     Assuming  k  =0,  determine  the  best 
speed,  best  vane  angle,  and  highest  efficiency. 

Solution. — The  best  speed  is  determined  from  equation  (9),  in  which 
c  =  1.2,  k=Q,  giving  u\/Vi  =.532.  From  equation  (11)  the  maximum 
efficiency  is  .951.  The  best  value  of  a\,  determined  from  equation  (12) 
with  the  above  value  of  Ui/Vi,  is  40°. 

2.  Solve  Ex.  1  assuming  k  =  .2. 

Ans.  WF,=.495;  e  =  .890;  ai=37°35'. 

3.  With  data  as  in  Ex.  1,  determine  what  value  of  k  reduces  the 
greatest  efficiency  to  .80,  and  find  the  corresponding  values  of  Ui/Vi 
and  a\. 

Solution. — This  must  be  solved  by  trial.  Trying  k  =  .5,  there  result 
the  values  u\/V\  =.456,  e  =  .820,  ai  =35°  15'.  Comparing  with  preceding 
results  it  may  be  estimated  that  the  required  value  of  k  is  about  .6. 
Solving  with  this  value,  Ui/V,  =.442,  e  =  .80,  a,  =34°  30'. 


193  THEORY   OF  IMPULSE  TURBINE. 

4.  Instead  of  the  assumption  w2  =  v2,  upon  which  the  above  solution 
for  maximum  efficiency  is  based,  it  is  sometimes  assumed  that  the  best 
velocity  is  that  which  makes  A2=90°.     Show  that  this  leads  to  an  equa- 
tion like  (9)  with  k'  substituted  for  k,  where  kf  =  (!  +  &)  sec2  a2  — 1. 

5.  Show  that  the  assumption  A?  =90°  leads  to  e=2(u\/V\)  cos  A\. 

6.  Take  data  as  in  Ex.  1  and  solve  on  the  assumption  A2=90°,  for 
fc=0,  .2,  and  .6.     Estimate  what  value  of  k  will  give  an  efficiency  of  .80. 

Ans.  For  k=  0,  W7,=.507,  e  =  .962,  a,  =38°  19'. 
"  fc  =  .2,  W7,=.471,e  =  .893,  a,=36°  5'. 
"  /b  =  .6,  wi/7,  =.422,  e  =  .798,  a,  =33°  28'. 

7.  Assume  dimensions  as  in  Ex.  1,  and  k  =  .6.     If  the  quantity  of 
water  to  be  utilized  is  16  cu.  ft.  per  sec.,  and  the  total  fall  is  600  ft.,  10 
per  cent  of  which  is  lost  by  friction  in  the  supply-pipe,  determine  the 
values  of  Ft  and  7i.        Ans.  7,  =  186.4.     Fi  =  16/186.4  =  .0858  sq.  ft. 

8.  In  Ex.  7  determine  highest  efficiency  and  power. 

Ans.  e  =  .80,  L  =432,000  ft.-lbs.  per  sec.     H.P.=785. 

9.  An  impulse  turbine  with  vertical  axis  and  axial  flow  is  to  utilize 
a  fall  of  10  ft.    Take  Zi-zz  =  l  ft.,  Ai  =26°,  a2  =  156°,  and  assume  that 
the  loss  of  head  in  the  supply-pipe  and  guide-passages   is  negligible. 
Determine  u/Vi,  e,  and  a\,  if  k=Q. 

Ans.  7i=25.4,  u  =  l5.5,  w/7i-.61,  e  =  .935,  a,  =56°  40'. 

10.  With  data  of  Ex.  9,  find  what  value  of  k  will  make  e  =  .80,  and 
determine  the  corresponding  values  of  u/V\  and  a\. 

Ans.  /c  =  .65,  w  =  13.1,  w/7,=.515,  e  =  .80,  a,  =48°  50'. 

11.  In  Ex.  10,  if  the  wheel  is  to  use  80  cu.  ft.  of  water  per  sec.,  what 
must  be  the  value  of  Fit    What  H.P.  will  be  realized? 


CHAPTER  XVIII. 
THE  TANGENTIAL  WATER  WHEEL. 

205.  Introductory  Illustration. — Let  Fig.  97  represent  a  sec- 
tion of  a  vane  or  bucket  receiving  a  jet  of  water  from  a  nozzle, 
and  let  the  vane  be  moving  in  the  same 
direction  as  the  jet,  the  velocity  of  the 


water  being  V\  and  that  of  the  vane 
u.  The  vane  consists  of  two  symmet- 
rical parts  coming  together  in  a  sharp 

edge.    This  edge  divides  the  jet  into  two  

streams  which  follow  the  two  parts  of  ! — ^-i 

the  vane,  being  gradually  deflected  and  ~~vT 

finally  discharged  at  its  outer  edges. 

The  force  exerted  upon  the  vane  by  the  jet.  and  the  work 
done  by  this  force  per  unit  time  may  be  computed  by  the 
methods  explained  in  Chapter  XIV.  With  the  notation  de- 
scribed in  Art.  174,  it  is  seen  that  U2  =  ui  =  u,  Ai=0,  ai=0, 
Vi  =  Vi—u,  while  a2  is  determined  by  the  tangent  to  the  vane 
curve  at  the  outer  edge. 

The  force  constantly  exerted  upon  the  vane  is  equal  to  the 
change  in  the  momentum  of  the  water  striking  it  in  one  second 
(Art.  158).  If  the  two  partial  streams  are  equal,  symmetry 
shows  that  the  direction  of  the  resultant  force  coincides  with 
that  in  which  the  vane  is  moving;  we  therefore  resolve  the 
momentum  in  this  direction.  The  momentum  of  W  Ibs.  of 
water  before  striking  the  vane  is  WVi/g,  while  that  of  an  equal 
quantity  leaving  the  vane  is  WVz/g.  The  component  of  the 
latter  in  the  direction  of  the  motion  of  the  vane  is 

W  W 

— V2  cos  A2  =  —  (u'+  v2  cos  a2). 
y  y 

199 


200  THE  TANGENTIAL   WATER  WHEEL. 

Hence  the  force  exerted  upon  the  vane  is 
W 

P  =  —  (Vi  —U—V2  COS  02). 

y 

If  frictional  resistance  be  neglected,  the  relative  velocity  will 
remain  constant  during  the  flow  over  the  vane,  so  that 


W 

and  P=  (1  -cos  a2)—  (7i  -  u). 

c/ 

The  work  done  per  second  by  the  force  P  is  L  =  Pu'}  or  on 
the  assumption  of  no  frictional  loss, 

W 

L=  (1  -cos  a2)  —  (Vi  —u)u. 
y 

For  a  given  value  of  W  this  work  has  its  greatest  value  when 
u  =  Vi/2,  the  value  being 

l-COSOa 

"" 


If  a2  =  180°,  this  reduces  to  WV?/2g,  showing  that  the  kinetic 
energy  of  the  water  striking  the  vane  is  all  given  up  to  the 
vane. 

206.  General  Features  of  Tangential  Water  Wheel.  —  The 

tangential  water  wheel  is  designed  to  realize  as  nearly  as  pos- 
sible the  ideal  conditions  assumed  in  the  foregoing  illustration. 
With  this  object  buckets  are  mounted  upon  the  rim  of  a  wheel, 
and  a  cylindrical  jet  is  thrown  against  these  in  such  a  way  that 
each  bucket,  while  receiving  the  jet,  is  moving  as  nearly  as 
practicable  in  the  same  direction  as  the  jet.  If  the  water  could 
be  received  and  gradually  deflected  by  each  bucket  without  dis- 
sipation of  energy  until  its  relative  velocity  became  equal  and 
opposite  to  the  velocity  of  the  bucket,  the  energy  of  the  jet 
would  be  wholly  utilized.  The  practical  limitations  which  pre- 


GENERAL  FEATURES  OF  TANGENTIAL  WATER  WHEEL.  201 

vent  a  close  approximation  to  this  ideal  case  are  indicated  in 
the  following  discussion. 

The  general  characteristics  of  wheels  of  this  type  are  shown 
in  Figs.  98  and  99.    The   fundamental  features  which  distin- 


FIG.  98.  Two  views  of  a  Doble  water  wheel  generating  8000  H.  P.  under 
a  working  head  of  1528  ft. 

guish  these  motors  from  those  of  the  Girard  type  are  the  use  of 
circular  nozzles  for  " guide-passages/'  and  the  double  or  " split  " 
character  of  the  buckets.  Tangential  wheel  buckets  have  been 
made  of  various  forms,  the  relative  merits  of  which  will  not  be 
discussed.  Separate  views  of  wheel-runners  of  two  different 
makes  are  represented  in  Figs.  100  and  101. 

The  relation  of  the  buckets  to  the  jet  is  shown  in  Fig.  102. 
Here  the  bucket  A  is  receiving  the  jet  centrally,  while  A'  is  just 


202 


THE  TANGENTIAL  WATER  WHEEL. 


entering  the  jet  and  'A"  is  nearly  in  the  position  where  the  jet 
ceases  to  strike  it.  While  receiving  the  jet  each  bucket  thus 
has  a  range  of  motion  A' A" ,  the  angle  between  its  direction  of 
motion  and  that  of  the  jet  varying  by  the  amount  A'CA". 
The  following  theoretical  discussion  will  refer  to  conditions  as 


FIG.  99.  Two  views  of  a  Pelton  unit  consisting  of  two  water  wheels  with 
a  total  capacity  of  7500  H.  P.  under  a  working  head  of  872  ft. 

they  exist  for  a  mean  position  of  the  bucket.  A  horizontal 
section  through  the  bucket  and  jet  in  this  mean  position  is 
shown  in  Fig.  102  (B).  The  motion  of  the  water  relative  to  the 
bucket  is  not,  however,  in  this  plane,  but  has  an  upward  com- 
ponent because  of  the  downward  component  of  the  velocity  of 
the  bucket.  This  is  shown  by  the  vector  triangle  at  (C).  The 
angle  a^  between  the  velocity  of  the  bucket  (u)  and  the  relative 


BEST  WHEEL  SPEED.  203 

velocity  of  the  water  leaving  the  wheel  (v2),  is  not  shown  in  its 
true  size  in  the  figure,  since  neither  u  nor  v2  is  parallel  to  the 
plane  of  the  sectional  view  (B)  .  A  mean  value,  depending  upon 
the  construction  of  the  bucket,  may  be  assumed  for  the  angle 
a2,  and  the  form  of  the  bucket  should  be  such  as  to  make  this 
angle  as  near  180°  as  will  permit  the  water  to  clear  the  following 
bucket. 

207.  Best  Wheel  Speed.  —  The  theory  of  the  tangential  water- 
wheel  may  be  expressed  in  substantially  the  same  form  as  that 
of  the  Girard  turbine,  if  the  same  notation  is  employed. 

The  relative  velocity  v\  of  the  water  just  about  to  strike 
the  bucket  is  computed,  for  a  given  value  of  u,  from  the  vector 
triangle  for  u,  i\,  V\.  This  triangle  lies  in  a  vertical  plane,  and 
is  shown  at  (C),  Fig.  102.  Thus 


(1) 


To  compute  v2  the  general  formula  (V)  *  is  to  be  applied; 
and  since  practically  u\  =  u2  and  z\  =  z2,  the  equation  becomes 


The  loss  of  head  hf  includes  the  loss  in  friction  of  the  stream 
while  flowing  over  the  bucket  ^surface,  the  loss  due  to  impinge- 
ment of  the  jet  against  the  sharp  edge  or  "splitter,"  and  the 
loss  due  to  the  interference  of  the  edge  of  the  bucket  with  the 
jet  before  it  is  in  position  to  receive  the  whole  stream.  These 
losses  all  vary  with  the  relative  velocity.  We  may  adopt  the 
usual  assumption  that  the  loss  is  proportional  to  the  square  of 
this  velocity,  and  express  it  in  the  same  way  as  in  the  theory  of 
the  Girard  turbine  : 


*Art.  179. 


204  THE  TANGENTIAL  WATER  WHEEL. 

Equations  (1),  (2)  and  (3)  then  give 


from  which  v2  may  be  found. 


FIG.  102. 


•     (4) 


The  next  step  would  be  to  substitute  in  the  general  formula 
(IV)  for  L 


cos 


cosa2, 


thus  expressing  L  in  terms  of  u.    The  resulting  equation  would 
be  identical  with  (7)  of  Chapter  XVII,  with  c  =  l. 

208.  Approximate  Solution  for  Best  Velocity.  —  The  appli- 
cation of  the  mathematical  condition  for  maximum  L  leads  to 
an  equation  of  the  fourth  degree  for  determining  u.  As  a 
simpler  and  sufficiently  exact  solution  the  same  approximate 


BEST  VELOCITY  —EFFICIENCY.  205 

assumption  may  be  made  as  in  the  case  of  the  Girard  turbine 
(Chapter  XVII). 

The  unutilized  energy  (per  pound  of  water)  is  made  up  of 
the  "frictional  "  loss  k(v22/2g)  and  the  kinetic  energy  possessed 
by  the  water  as  it  leaves  the  wheel,  V22/2g.  While  the  former 
is  the  more  important,  the  latter  probably  varies  more  rapidly 
as  the  speed  varies  from  that  giving  greatest  efficiency,  so  that 
the  highest  efficiency  will  be  obtained  when  V2  is  small,  and 
this  will  be  when  v2  and  u  are  nearly  equal.  Assuming 

v2  =  u,    ........     (5) 

equation  (4)  becomes 


0,    ....    (6) 
from  which  u/V\  may  be  computed. 

209.  Efficiency.  —  Taking  as  the  available  energy  the  kinetic 
energy  of  the  jet  from  the  nozzle,  the  hydraulic  efficiency  is 


Putting  V2  =  u,  s2  =  u  +  v2  cos  a2  =  w(l+cos  a2),  the  highest  effi- 
ciency according  to  the  foregoing  solution  is 

[u  v?  "1 

y-cosAi-(l+cosa2)y-£^,     ...     (8) 

in  which  u/Vi  must  have  the  value  given  by  equation  (6)  . 

210.  Form  of  Bucket  Surface.  —  While  a  bucket  is  receiving 
the  jet  it  moves  through  a  certain  distance  in  a  direction  oblique 
to  that  of  the  jet  and  turns  through  a  small  angle.  The  point 
at  which  the  jet  strikes  it,  and  the  direction  of  the  relative 
motion  of  the  impinging  water,  are  therefore  variable,  and  the 
form  of  the  bucket  surface  should  conform  to  this  variation. 
Without  entering  into  a  detailed  study,  it  may  be  said  that  the 
form  should  be  such  as  to  deflect  the  water  as  gradually  as  pos- 
sible throughout  its  entire  passage  over  the  bucket  surface, 
and  to  make  the  direction  of  the  relative  velocity  of  outflow  as 


206  THE  TANGENTIAL  WATER  WHEEL. 

nearly  opposite  to  the  velocity  of  the  bucket  as  practicable. 
The  varying  aspect  presented  by  the  bucket  to  the  stream 
seems  to  require  a  surface  of  double  curvature  if  sudden  changes 
of  velocity  are  to  be  avoided.  The  dividing  edge  should,  of 
course,  be  as  sharp  as  practicable.  The  limitation  of  the  direc- 
tion of  outflow  imposed  by  the  necessity  that  the  water  leaving 
one  bucket  shall  clear  the  next  one  has  already  been  mentioned. 

211.  Comparison  with  Theory  of  Girard  Turbine. — The  for- 
mulas above  deduced  for  the  tangential  water  wheel  are  seen 
to  be  identical  in  form  with  those  obtained  for  the  radial-flow 
Girard  turbine  except  that  c  =  l.  They  may  also  be  obtained 
from  the  formulas  applying  to  the  axial-flow  Girard  wheel  by 
putting  Zi=z2. 

212*  Conditions  Favorable  for  Use  of  Tangential  Water 
Wheel. — In  general  it  may  be  said  that  wheels  of  this  type  may 
be  used  to  advantage  wherever  the  fall  utilized  is  great  and 
the  supply  of  water  relatively  small,  and  are  at  a  decided  dis- 
advantage only  when  the  fall  is  but  a  few  feet,  or  when  the 
supply  of  water  is  great.  They  are  in  efficient  operation  under 
falls  as  great  as  2500  ft. 

213.  Actual  Efficiencies. — While  accurate  tests  are  rare,  it 
is  probable  that  the   best   tangential  water  wheels  give  effi- 
ciencies as  great  as  are  obtained  from  any  type  of  turbine  or 
water  wheel.     Hydraulic  efficiencies  as  great  as  80  to  85  per 
cent  under  heads  up  to   1500  or  2000  ft.  are  doubtless  not 
uncommon.    The  extremely  high  efficiencies  sometimes  claimed 
by  manufacturers  must  be  regarded  with  suspicion. 

214.  Regulation. — An    important    problem    in    connection 
with  the  practical  working  of  a  water  wheel  is  the  regulation  of 
the  discharge  from  the  nozzle  in  order  to  vary  the  power  output 
of  the  wheel,  or  to  conform  to  variations  in  the  supply  of  water. 
The  partial  closing  of  a  valve  of  any  ordinary  form  placed  in 
the  supply-pipe  or  nozzle  would  cause  a  serious  loss  of  energy. 
In  some  cases  two  or  more  nozzles  are  placed  at  different  points 


GOVERNING.  207 

of  the  circumference  of  the  wheel,  any  combination  of  which 
may  be  used  as  required. 

The  needle  nozzle  (Fig.  103)  is  a  regulating  nozzle  of  peculiar 


FIG.  103. 

form,  designed  to  vary  the  size  of  the  jet  at  pleasure  with  little 
loss  of  efficiency.  The  conical  valve  or  "needle  "  can  be  moved 
parallel  to  the  axis  of  the  jet,  so  as  to  leave  any  desired  amount 
of  opening  between  the  needle  and  the  nozzle  tip.  If  carefully 
made,  this  device  accomplishes  the  object  of  regulation  with 
little  loss  of  energy  very  satisfactorily.* 

215.  Governing. — Another  important  problem  is  that  of 
maintaining  a  uniform  speed  of  rotation  when  the  load  fluctu- 
ates. This  is  usually  accomplished  by  deflecting  the  jet  so  that 
a  varying  portion  of  it  strikes  the  buckets,  the  deflection  being 
permitted  by  a  joint  in  the  nozzle  pipe,  and  the  movement 
being  controlled  by  some  form  of  centrifugal  governor. 

Governing  has  also  been  successfully  accomplished  by  means 
of  the  needle  nozzle,  the  needle  being  actuated  by  the  governor 
so  that  the  nozzle  opening  varies  with  any  variation  in  the 
speed. 

EXAMPLES. 

1.  If  Ai  =20°  and  a2  =  160°,  determine  best  value  of  u/Vi  and  highest 
efficiency  assuming  /c=0.  Ans.  u/V\  =.532.     e  =  .966. 

2.  Solve  Ex.  1  assuming  k  =  .5.  Ans.  u/V  =  A73.     e  =  .860. 

3.  In  the  same  case,  if  the  maximum  efficiency  is  .80,  what  is  the 
value  of  k,  and  what  is  the  best  wheel  velocity? 

4.  If  the  effective  head  at  the  nozzle  is  600  ft.  and  the  wheel  is  to 
make  700  R.P.M.,  what  should  be  its  diameter? 

5.  If  the  nozzle  throws  a  jet  1.5  inches  in  diameter,  determine  the 
power. 

*  The  needle  nozzle  is  covered  by  U.  S.  patent. 


CHAPTER  XIX. 


THEORY  OF  THE  REACTION  TURBINE. 

216.  General  Features  of  Reaction  Turbines. — The  funda- 
mental characteristic  of  a  reaction  turbine  is  the  fact  that 

the  wheel  passages  are  completely  filled 
by  the  streams  flowing  through  them. 
The  pressure  within  these  passages, 
therefore,  is  not  determined  by  contact 
of  the  stream  with  air  as  in  the  impulse 
wheel,  but  in  general  varies  continuously 
between  the  points  of  admission  and 
discharge. 

The  general  arrangement  of  a  reac- 
tion turbine  with  radial  outward  flow 
(the  Fourneyron  type)  is  shown  in  Fig. 
92,  while  the  case  of  axial  flow  (Jonval) 
is  represented  in  Fig.  104,  and  that  of 
inward  flow  (Francis  type)  in  Fig.  105. 
The  turbine  shown  in  Fig.  105  does  not  closely  resemble  the 
original  design  of  Francis,  being  adapted  to  a  much  greater 
fall. 

A  design  quite  generally  adopted  in  American  practice  is 
that  of  mixed  flow  (inward  admission  and  axial  discharge). 
This  is  represented  in  Fig.  106.  There  is  no  sharp  distinction 
between  this  and  the  form  represented  in  Fig.  105. 

Turbine  runners  of  two  forms  are  shown  in  Figs.  107 
and  108,  the  former  being  of  the  inward-flow  or  Francis 
type,  the  latter  of  the  mixed-flow  type  similar  to  those  in 
Fig.  106. 

208 


FIG.  104. 


FIG.  100. 
PELTON  WATER-WHEEL  RUNNER. 


FIG.  101. 
DOBLE  WATER-WHEEL  RUNNER. 


FIG.  107. 

RUNNER  OF  FRANCIS  TURBINE 
(PLATT  IRON  WORKS  COMPANY). 


FIG.  108. 

RUNNER  OF  VICTOR  TURBINE 
(PLATT  IRON  WORKS  COMPANY). 


GENERAL  FEATURES  OF  REACTION  TURBINES.         209 


FIG.  105.     Modern  Francis  turbine,  of  capacity  5000  H.  P.,  under  a  fall  of 
320  ft.     Built  by  the  Allis-Chalmers  Company. 


FIG.  106.  A  pair  of  turbines  installed  by  the  I.  P.  Morris  Co.  The  work, 
ing  head  is  82  ft.,  the  speed  600  revolutions  per  minute,  and  the  power  gener- 
ated 750  H.P. 


210  THEORY   OF  THE  REACTION  TURBINE. 

The  following  theory  as  at  first  presented  refers  to  the  Four. 
neyron  or  outward-  flow  turbine,  as  represented  in  Fig.  92.  The 
main  features  of  the  theory  hold  for  reaction  turbines  of  other 
forms,  the  points  of  difference  being  indicated  later. 

217.  Data  and  Notation.  —  In  the  following  discussion  of 
reaction  turbines  the  notation  will  for  the  most  part  be  the 
same  used  in  the  foregoing  theory  of  impulse  wheels  and  ex- 
plained in  Art.  174.  In  addition  the  following  symbols  will  be 
used. 

Assuming  the  wheel  to  discharge  into  the  atmosphere,*  let 
h  denote  the  total  fall  from  surface  of  supply  reservoir  or  head- 
race to  the  place  of  discharge  from  the  wheel,  and  let  h'  denote 
the  total  loss  of  head  (i.e.,  the  part  of  h  that  is  not  utilized). 

In  applying  to  the  reaction  turbine  the  general  formulas 
deduced  in  Chapter  XV,  the  special  condition  must  be  intro- 
duced that  the  cross-section  of  the  stream  within  the  wheel  is 
everywhere  fixed,  not  varying  with  the  wheel  speed  nor  with  the 
velocity  of  flow.  Thus  in  the  equation  of  continuity  (formula 
(I'),  Art.  179)  both  FI  and  /2  are  constant,  so  that  v%  and  V\ 
are  in  a  constant  ratio.  That  is, 


........    (1) 

in  which  c'  =  FI  //2  =  constant  .f 

The  following  data  will-  be  taken  as  known:  h,  A\,  a^  c?  = 
Fi/f'2,  c  =  r2/ri. 

218.  Relation  between  Wheel  Speed  and  Rate  of  Flow  through 
Wheel.  —  The  velocity  of  flow  in  every  section  of  the  stream  is 

*  If  the  wheel  is  submerged,  h  will  mean  the  fall  from  surface  of  supply 
water  to  surface  of  waste  water;  the  same  formulas  will  then  apply  as  in 
the  case  of  discharge  into  the  air.  The  effect  of  a  suction-tube  is  considered 
in  Art.  228. 

fit  will  be  noticed  that  the  condition  Fi//a=  constant  in  the  reaction 
turbine  replaces  the  condition  p\  =  pz  in  the  impulse  turbine.  Formula  (V), 
which  was  used  in  determining  the  flow  through  the  wheel  in  the  previous 
case,  is  of  use  in  the  present  problem  only  in  determining  the  relation  between 
pl  and  p2. 


RATE  OF  FLOW  THROUGH  WHEEL.  211 

determined  if  that  in  any  one  section  is  fixed.  Let  all  such 
velocities  be  expressed  in  terms  of  FI,  and  let  it  be  required  to 
determine  the  relation  between  Vi  and  the  wheel  speed  HI. 

From  the  above  meanings  of  h  and  h'  it  is  seen  that  h  —  h' 
is  the  utilized  head,  and  that  the  energy  received  by  the  wheel 
from  the  water  per  unit  time  is 

L  =  W(h-h') (2) 

Hence  from  formula  (IV),  Art.  179, 

W 

W(h—h')=  —  (t*i$j[  —  U2S2) , 

tj 

or  g(h—h')=UiSi—u2s2 (3) 

The  second  member  of  this  equation  can  be  expressed  in  terms 
of  YI  and  u\.  Thus 

u2  =  cui ;    Si  =  Vi  cos  AI  ; 
S2  =  u2  +  v2  cos  a2  =  c 
so  that  (3)  may  be  written 

g(h  —  h')  =  (cosAi—cc?  cosa2)ViUL  —  c2u-f.      .    .     (4) 

It  is  now  necessary  to  express  the  value  of  h'. 

The  most  important  losses  of  head  may  be  expressed  by  two 
terms.  One  of  these  represents  the  "frictional "  loss  occurring 
throughout  the  stream,  which  by  the  usual  rule  of  hydraulics  is 
expressed  as  proportional  to  the  square  of  the  velocity  of  flow. 
Since  the  velocities  in  all  sections  vary  in  the  same  ratio,  we 
may  assume 

o 

k-~-  =  total  frictional  loss  of  head, 
^9 

k  being  a  coefficient  whose  value  depends  upon  the  dimensions 
and  character  of  the  entire  series  of  passages,  and  which  will 


212  THEORY  OF  THE   REACTION  TURBINE. 

be  treated  as  constant.  The  other  important  loss  of  head  is 
that  represented  by  the  kinetic  energy  of  the  water  leaving 
the  wheel.  Combining  this  with  the  frictional  loss,  we  may 
write 


» 


From  the  vector  triangle  for  u2,  v2,  and  F2, 

V22  =  U22  +  V22  +  2U2V2  COS  a2 

=  c2ul2  +  c'2V12+2cc'ulVl  cos  a2, 
so  that  (5)  may  be  written  in  the  form 

2gh'  =  (1  +  k)  c'WJ  +  2ccf  cos  a2  •  Vlul  +  c2uj.  .    .    (6) 
Eliminating  h'  between  (4)  and  (6), 

.    .     (7) 


By  solving  this  equation  Vi  may  be  determined  for  any 
value  of  MI.  For  a  given  wheel,  FI  being  fixed  in  value,  the 
rate  of  discharge  varies  as  V\,  being  given  (when  V\  is  known) 
by  the  equation 

W^wFtVi.     .......     (8) 

219.  Power  and  Efficiency  for  any  Wheel  Velocity.  —  Ex- 

pressing UiSi—  u2s2  in  terms  of  u\  and  V\  as  in  Art.  218,  the 
general  formula  (IV)  may  be  written 

W 

L=—  [(cos  AI  -cc'  cos  a2)ViUi  -c2Ui2].  ...     (9) 

From  this,  by  using  the  value  of  FI  given  by  (7),  the  power 
corresponding  to  any  value  of  u\  may  be  computed,  provided 
FI  is  known  so  that  W  can  be  determined. 

The  efficiency,  however,  is  independent  of  the  value  of  FI. 
For  W  Ibs.  of  water  the  available  energy  is  Wh  foot-pounds, 
hence 

e  =  rr  =  ~[(cos  AI  -  cc'  cos  a2)  ViUi  -  c%i2J,      .     (10) 


BEST  SPEED  AND  HIGHEST  EFFICIENCY. 

in  which  V\  must  have  the  value  given  by  (7)  for  any  value 
f 


220.  Best  Speed  and  Highest  Efficiency.  —  The  speed  giving 
maximum  efficiency  may  be  determined  by  applying  the  con- 
dition de/dui=Q.  The  direct  method  of  procedure  would  be 
to  solve  equation  (7)  for  V\  and  substitute  its  value  in  (10), 
thus  expressing  e  explicitly  in  terms  of  the  single  variable  HI 
before  differentiating.  The  following  indirect  method  is  less 
laborious. 

Differentiating  (10), 

.    de      .  .     dV, 

gh  ~-j  —  =  (cos  AI  —  cc  cos  a2)Ui—j— 


+  (cos  AI  -cc'  cos  a2)Vi  —  2c2Wi  =0. 
Differentiating  (7), 

+  cos  AI  •  wj-r^-f  cos  ^i  -  7i  -c2wi  =0. 


Eliminating  dVi/dui  between  these  two  equations,  and  reduc- 
ing, 

(l+&)c'2(cos  Ai-ccf  cos  a2)Vl2-2(l+k)c2cf2Vlu1 

-c2(cos  AI  +ccr  cos  a2)wi2=0.      (11) 

This  equation  must  be  satisfied  when  e  is  a  maximum.  Com- 
bining it  with  (7)  ,  which  is  always  true,  the  values  of  u\  and  V\ 
corresponding  to  maximum  efficiency  may  be  found.  The 
solution  may  be  expressed  as  follows  : 

Equation  (11)  determines  *  the  value  of  Vi/u\.    Calling  this 
a,  and  substituting  Vi  =  aui  in  (7), 

[(!+A;)cr2a2  +  2cosA1.a:-c2]wi2=2^,    .     .     .     (12) 

from  which  may  be  computed  the  value  of  u\  giving  maximum 
efficiency. 

*  Two  values  of  V\/u\  are  given  by  (11);  which  value  corresponds  to  the 
practical  problem  is  seen  in  any  particular  case  after  substitution  in  (7). 


214  THEORY  OF  THE  REACTION  TURBINE. 

221.  Best  Angle  of  Wheel  Vane.  —  The  direction  of  the  wheel 
.vane  at  the  point  of  inflow  should  agree  with  that  of  the  rela- 
tive velocity  of  outflow  from  the  guide  passages,  in  order  to 
prevent  sudden  deflection  of  the  water  and  consequent  loss  of 
energy.  The  value  of  ai  is  fixed  by  that  of  VI/MI,  and  may 
be  determined  in  the  usual  manner  by  solving  the  vector  tri- 
angle whose  sides  are  MI,  Vi,  Vi.  Equation  (12)  of  Art.  200 
may  be  employed  : 


i.     .    .    .     (13) 

Since  u\/V\  varies  with  the  wheel  velocity,  ai  also  varies  with 
u\.  The  value  corresponding  to  maximum  efficiency  should 
govern  the  design  of  the  wheel. 

In  the  foregoing  theory  the  loss  of  head  due  to  sudden 
deflection  of  the  stream  entering  the  wheel  is  neglected,  which 
is  equivalent  to  assuming  that  the  vane  angle  varies  with  the 
speed  of  rotation  so  as  always  to  agree  with  ai.  This  is  allow- 
able in  solving  for  maximum  efficiency,  since  the  vane  angle 
is  to  be  made  to  agree  with  the  value  of  ai  which  is  finally 
found  as  the  result  of  the  solution. 

For  a  given  wheel,  however,  there  is  only  a  single  value  of 
Ui/Vi  which  makes  the  assumption  of  no  sudden  deflection  of 
the  stream  true,  and  the  loss  of  head  due  to  this  cause  should 
be  taken  into  account  in  an  accurate  solution  of  the  problem  of 
flow  through  the  wheel  for  any  rotation  speed.  In  spite  of 
this,  however,  it  is  probable  that  equation  (7)  shows  approxi- 
mately the  way  in  which  V\  varies  with  Ui  in  an  actual  wheel, 
for  a  considerable  range  of  values  of  MI. 

222.  Pressure  at  Entrance  to  Wheel.—  If  hi  denotes  the  fall 
from  head-race  to  point  of  outflow  from  guides,  and  Hf  the 
loss  of  'head  between  these  points,  the  equation  of  energy  gives 


pi  being  reckoned  from  atmospheric  pressure  as  zero. 


INTRODUCTION    OF  RATIOS  OF  VELOCITIES.  215 

EXAMPLES. 

Given  A^  =28°,  a,  =  158°,  n  =3.375  ft.,  r2  =4.146  ft.,  F,  =6.537  sq.  ft., 
/8=  7.687  sq.  ft.,  h  =  12.8  ft. 

1.  Assuming   fc  =  .5,  determine  best  wheel  velocity,  best  value  of 
vane  angle,  and  highest  efficiency. 

Ans.  w,=.50\/2^=28.7ft.  per  sec.;    e  =  .71;    a,  =61°  45'. 

2.  By  trial  of  different  values  of  k,  determine  what  value  gives  a 
maximum  efficiency  of  80  per  cent.  Ans.  A;  =  .25  very  nearly. 

223.  Introduction  of  Ratios  of  Velocities.  —  Let  m  denote  the 
velocity  equivalent  to  the  fall  h,  i.e., 


then  it  is  seen  that  the  main  equations  reached  by  the  above 
theory  really  involve  the  ratios  of  the  three  velocities  Ui,  Vt,  m, 
rather  than  their  actual  values.  Thus,  comparing  different 
cases  in  which  h  has  different  values,  if  HI,  Vi  and  m  are  changed 
in  the  same  ratio,  equations  (7),  (10),  and  (11)  are  unchanged* 
It  is  instructive  to  write  the  equations  so  as  to  involve  the 
ratios  of  the  three  velocities  explicitly. 

Let  Ui/m=x,  Vi/m  =  y.    Then  equations  (7),  (10),  and  (11) 
become 

osAl-yx-c2x2  =  l,     ...    (A) 


e  =  2  (cos  A  i  -  cc'  cos  o2)  yx  -  2c2x2,     .    .    .     (B) 

(1  +&X2(coB  AI  -cc'  cos  a2)y2  -2(1  +k)c2c'2yx 

-  c2(cos  AI  +cc*  cos  a2).x2=0.     (C) 

Of  these  equations  (A)  and  (B)  are  general,  while  (C)  holds  only 
for  maximum  efficiency. 

The  values  of  the  rate  of  discharge  and  of  the  power  depend 
not  merely  upon  ratios  of  velocities,  but  upon  their  actual 
values.  They  may  be  written  as  follows: 


(D) 


—^  —  ey.  .    .    .    (E) 


216 


THEORY  OF  THE  REACTION  TURBINE. 


224.  Graphical  Representation. — If  x  and  y  are  taken  as 
rectangular  coordinates,  equation  (A)  represents  a  curve  which 
shows  graphically  the  way  in  which  the  rate  of  discharge  varies 
with  the  wheel  velocity  for  any  constant  fall;  for  when  m  in 
constant,  Vi  is  proportional  to  y  and  HI  to  x. 

The  value  of  e  for  any  value  of  x  and  the  corresponding 
value  of  y  may  be  computed  from  (B).  If  a  curve  be  drawn 
with  e  as  ordinate  and  x  as  abscissa,  it  shows  graphically  the 


1.00 


.80 


.60 


.40 


.20 


(E) 


20    Values    .40  .60 

FIG,  109. 


.80 


1.00 


variation  of  the  efficiency  with  the  wheel  speed  when  h  is  con- 
stant. 

The  variation  of  the  power  with  the  wheel  velocity  when  h 
is  constant  is  shown  by  a  curve  determined  by  taking  as  abscissas 
values  of  x  and  as  ordinates  values  of  ey,  the  variable  factor 
in  the  value  of  L  as  given  by  (E) . 

These  three  curves,  for  the  data  of  the  examples  after  Art. 
222,  with  A; =0.25,  are  shown  in  Fig.  109. 


APPROXIMATE  SOLUTION  FOR  MAXIMUM  EFFICIENCY.     217 

Equation  (A)  represents  an  hyperbola,  while  (C)  represents 
two  straight  lines,  one  of  which  intersects  the  hyperbola  in  a 
point  whose  coordinates  are  the  values  of  x  and  y  for  maximum 
efficiency. 

These  curves  cannot,  of  course,  be  regarded  as  accurately 
representing  the  results  actually  to  be  expected  from  a  turbine 
of  given  dimensions.  The  uncertain  element  in  the  theory  is 
the  value  of  the  loss  of  head  h'  ';  in  particular  the  remarks  made 
in  Art.  221  regarding  loss  at  entrance  must  be  borne  in  mind. 

It  should  be  noticed  that  the  value  of  e  given  by  equation 
(B)  involves  no  constants  except  dimensions  of  wheel  and 
guides,  and  that  this  equation  is  not  affected  by  the  uncer- 
tainty in  the  value  of  hf.  If  the  true  relation  between  x  and  y 
were  known  (as,  for  example,  by  experiment),  the  efficiency 
curve  could  be  correctly  drawn  from  equation  (B). 

225.  Approximate    Solution     for     Maximum     Efficiency.— 

Reasoning  as  in  the  case  of  the  impulse  turbine  (Art.  198),  it 
appears  that  the  greatest  efficiency  will  result  when  the  absolute 
velocity  of  outflow  from  the  wheel  has  a  small  value.  The 
assumption  ^2  =  ^2  will  therefore  give  an  approximation  to  the 
solution  for  maximum  efficiency.  The  more  common  assump- 
tion, however,  is  that  ^4.2  =  90°,  or 

V2  cosA2  =  u2  +  V2  cos  a2-=0  .....     (14) 


Either  of  these  assumptions  leads  to  a  simple  equation  in  place 
of  (C).    Thus,  from  (14), 


or  cx+c'  cos  a2-2/=0,    ......    (C') 

which  replaces  (C).    The  solution  is  otherwise  unchanged. 

In  the  case  represented  by  the  curves  in  Fig.  107  the  ap- 
proximate solution  for  maximum  efficiency  gives  practically  the 
same  result  as  the  exact  solution.  The  value  of  y/x  or  a  given 
by  (C')  is  almost  identical  with  that  given  by  (C);  both  these 
equations  are  represented  by  the  straight  line  marked  (C),  and 


218  THEORY   OF  THE  REACTION  TURBINE. 

the  intersection  of  this  line  with  the  curve  (A)  gives  the  values 
of  x  and  y  corresponding  to  maximum  efficiency. 

EXAMPLES. 

1.  Using  the  results  of  the  exact  solution  of  Ex.  1,  Art.  222,  deter 
mine  the  value  of  A2.  Ans.  92°  30'. 

2.  With  same  data,  assume  ^4.2=90°,  and  determine  x,  y,  e,  ai. 

Ans.  3  =  .51,  2/  =  .79,  e  =  .71,  a,=Q2°  50'. 

226.  Cases  of  Inward,  Axial,  and  Mixed  Flow.— All  these 
cases  are  covered  by  the  foregoing  theory,  so  long  as  the  wheel 
discharges  directly  into  the  air  or  below  the  surface  of  the  tail- 
race.  With  axial  flow  c  =  l,  with  inward  flow  or  mixed  inward 
and  axial  flow  c<l;  but  the  same  general  formulas  apply  to 
all  cases  of  flow.  There  is,  however,  a  particular  limitation  to 
the  accuracy  of  the  formulas  as  applied  to  the  cases  of  axial 
and  mixed  flow. 

In  deriving  the  formulas  it  is  assumed  that  r\  and  r2  have 
the  same  values  for  all  particles.  This  is  strictly  true  in  the 
Fourneyron  type  of  turbine  (Fig.  92),  and  approximately  true 
in  the  Francis  (Fig.  105).  In  the  Jonval  (Fig.  104)  it  is  not 
true;  but  since  the  variation  in  the  value  of  r  for  different  par- 
ticles is  relatively  small,  it  is  fairly  satisfactory  in  this  case  to 
use  an  average  value  as  applying  to  all  particles. 

In  the  case  of  mixed  flow  represented  in  Fig.  106,  the  radius 
of  admission  r±  is  the  same  for  all  particles,  but  the  range  of 
values  of  r2  for  different  particles  is  very  great.  By  using  a 
mean  value  of  r%  the  formulas  will  still  apply  roughly.  With 
any  given  wheel  it  will  doubtless  be  approximately  true  that 
the  highest  efficiency  results  when  the  speed  is  such  that  the 
total  angular  momentum  of  the  water  leaving  the  runner  is 
zero,  but  it  does  not  seem  possible  to  get  a  useful  expression 
for  the  value  of  this  angular  momentum  in  terms  of  the  con- 
stants and  variables  of  the  problem.  The  actual  direction  of 
motion  of  a  particle  leaving  the  runner  at  any  point  is  uncertain 
even  when  the  form  of  the  runner  is  given,  and  theory  cannot 
furnish  definite  rules  for  the  design  of  the  vanes. 


CASE  OF  DISCHARGE  INTO  DIVERGING   PASSAGES.       219 


227.  Case  of  Discharge  into  Diverging  Passages. — If  the 
wheel  discharges  into  passages  which  diverge  so  as  gradually 
to  reduce  the  velocity  before  the  stream  passes  into  the  air 
or  into  a  body  of  free  water,  the  kinetic  energy  represented  by 
the  absolute  velocity  of  outflow  from  the  wheel  may  not  be 
wholly  lost. 

Fig.  110  represents  a  "diffuser,"  devised  by  Boy  den,  used 
with  a  submerged  wheel  with, outward  flow.  The  vertical  sec- 


FIG.  no. 

tion  shows  that  the  diffuser  acts  practically  as  a  diverging 
tube,  receiving  the  water  from  the  wheel  with  absolute  velocity 
V2  and  discharging  it  with  a  less  velocity.  Experiments  by 
Francis  *  showed  a  gain  of  three  per  cent  in  the  efficiency  by 
the  use  of  the  diffuser. 

To  take  account  of  the  possible  saving  of  energy  due  to 
discharge  into  diverging  passages,  the  theory  may  be  modified 
by  replacing  the  term  V22/2g  in  the  value  of  h'  by  k'(V22/2g), 
k'  being  a  fraction  which  may  be  assumed  constant  for  the  pur- 
pose of  the  theory.  Carrying  out  the  solution  as  before,  the 
equation  which  replaces  (A)  is 

(k+k')c'2y2+2[cos  Ai~(l-k')cc' cos  a2]yx-(2-k')c2x2=l.  (A') 
No  change  is  made  in  equation  (B) : 

e  =  2 (cos  Ai-cc'  cos  a2) yx  - 2c2x2.      .    .     .     (B') 

The  equation  replacing  (C) ,  given  by  the  condition  de/dx  =  0, 
may  be  found  without  difficulty,  but  we  will  here  use  instead 
the  equation  resulting  from  the  assumption  A2  =  90°,  which  is 
the  same  as  above  given : 

'cx  +  cf  cosa2-i/  =  0.      .....     (C') 

*  Lowell  Hydraulic  Experiments,  p,  5. 


220  THEORY  OF  THE  REACTION  TURBINE. 

228.  Effect  of  Suction  Tube.  —  If  the  wheel  discharges  into 
a  suction  tube,  the  kinetic  energy  possessed  by  the  water  as  it 
leaves  the  buckets  may  not  be  wholly  lost.  This  will  be  true 
if  the  construction  is  such  that  the  change  of  velocity  of  the 
stream  in  passing  from  the  buckets  into  the  tube  is  gradual. 
The  energy  corresponding  to  the  velocity  of  outflow  from  the 
suction  tube  into  the  tail  race  will,  however,  be  lost.  It  would 
seem  that  a  saving  might  on  the  whole  be  effected  by  the  use  of 
a  diverging  tube,  which  should  receive  the  water  from  the  wheel 
with  as  little  change  of  velocity  as  possible,  and  gradually  reduce 
the  velocity  before  discharging  the  stream  into  the  tail  race. 

While  it  is  not  possible  to  express  with  any  exactness  the 
value  of  the  loss  of  head  in  this  case,  the  assumption  made  in 
Art.  227  will  serve  as  a  basis  for  an  approximate  solution  of 
the  problem  of  design  for  maximum  efficiency.  That  is,  it  may 
be  assumed  that 


in  which  kf  lies  between  0  and  1.  If  it  be  assumed  also  that 
the  best  speed  is  that  which  makes  the  tangential  component 
of  V2  zero,  the  resulting  equations  are  identical  with  those  of 
Art.  227. 

In  this  case  there  is  a  special  reason  for  imposing  the  con- 
dition ^2  =  0,  for  if  this  is  not  satisfied  the  water  enters  the 
suction  tube  with  a  whirling  motion  which  must  increase  the 
loss  of  energy. 

The  pressure  at  any  point  within  the  suction  tube  depends 
upon  the  height  above  the  tail  water,  the  velocity  of  flow,  and 
the  loss  of  head  in  the  tube.  Thus  let  the  equation  of  energy 
be  written  for  the  sections  Y  and  Z  (Fig.  104)  .  If  z,  p,  v  refer 
to  the  point  Y,  po  =  atmospheric  pressure,  and  H'  =  loss  of  head 
between  Y  and  Zf 


REGULATION  AND  GOVERNING    ACTUAL  EFFICIENCIES.  221 


w 


---a- 

w     w  g 

229.  Regulation  and  Governing.  —  The  problem  of  regulat- 
ing the  quantity  of  water  supplied  to  the  wheel,  without  serious 
loss  of  efficiency,  is  more  difficult  with  reaction  turbines  than 
with  those  of  the  impulse  type.     With  the  latter  it  is  only  nec- 
essary to  vary  the  cross-section  of  the  guide  passages  at  the  place 
of  discharge,  and  this  can  be  done  with  relatively  little  loss  of 
energy.     With   a  reaction  turbine,  however,  since  the  wheel- 
passages  are  always  filled,  a  throttling  of  the  stream  at  any 
point  causes  not  only  a  contraction  but  a  subsequent  expansion 
of  the  stream,  resulting  in  serious  loss  of  energy.     Perhaps  the 
most  common  regulating  device  is  a  cylindrical  gate  fitting  over 
the  guide  openings,  which  can  be  adjusted  to  any  desired  amount 
of  opening.     With  such  a  gate  a  turbine  can  give  its  highest 
efficiency  only  when  the  gate  is  fully  open. 

Regulation  by  pivoted  guide-vanes  was  introduced  by  Prof. 
James  Thomson,  with  the  inward-flow  turbine  designed  by  him. 
The  same  device  has  been  used  with  modern  American  turbines 
of  the  Francis  type.  By  simultaneous  rotation  of  the  pivoted 
vanes  the  area  of  the  guide  openings  may  be  varied  with  little 
or  no  loss  of  energy.  The  angle  A\  also  changes,  but  without 
important  effect  on  the  efficiency. 

Governing  also  is  accomplished  in  a  satisfactory  manner  by 
means  of  the  pivoted  guides,  these  being  actuated  by  some 
form  of  centrifugal  governor. 

230.  Actual  Efficiencies  of  Reaction  Turbines.  —  The  diffi- 
culty of  making  accurate  efficiency  tests  is  so  great  that  com- 
paratively few  reliable  data  exist  regarding  actual  efficiencies 
of  turbines  under  practical  working  conditions.     Such  a  test 
requires  the  accurate  measurement  of  the  quantity  of  water 
used  per  second,  of  the  fall,  and  of  the  power  output  of  the 
turbine.     Laboratory  tests  upon  full-sized  machines  are  gener- 


THEORY  OF  THE  REACTION  TURBINE. 

ally  out  of  the  question,  and  power  plants  in  actual  operation 
are  rarely  so  arranged  that  the  requisite  measurements  can  be 
made  with  a  good  degree  of  accuracy.  It  must  be  remembered 
that  in  the  foregoing  theory  it  is  the  hydraulic  efficiency  that 
is  referred  to,  and  that  this  cannot  be  determined  experiment- 
ally; it  is  necessary  to  measure  the  power  obtained  from  the 
motor  rather  than  that  given  up  to  it  by  the  water.  And  in 
most  cases  this  power  output  has  to  be  measured  electrically, 
thus  introducing  electrical  losses  of  more  or  less  uncertain  value 
as  well  as  errors  due  to  the  electrical  measuring  instruments. 

The  experiments  of  Francis  at  Lowell  *  upon  an  outward- 
flow  (Fourneyron  type)  turbine  under  a  fall  of  about  13 
feet,  gave  an  efficiency  at  full  gate  of  79  per  cent.  In  this 
case  the  power  output  was  measured  by  means  of  a  friction 
brake,  so  that  no  electrical  losses  or  uncertainties  were  involved. 
It  is  claimed  that  some  of  the  best  modern  turbines  give  effi- 
ciencies as  high  as  87  per  cent,  and  it  is  probable  that  hydraulic 
efficiencies  as  high  as  85  per  cent  are  often  realized  by  well 
designed  turbines  when  working  with  full  gate  opening.  With 
ordinary  methods  of  regulation  the  partial  closing  of  the  gate 
causes  an  important  decrease  in  the  efficiency.  Regulation  by 
pivoted  guides,  with  the  object  of  avoiding  such  decrease,  has 
already  been  referred  to  (Art.  229). 

Another  important  consideration  affecting  efficient  working 
is  the  relation  of  wheel  speed  to  fall.  The  above  theory  has 
made  it  clear  that  a  turbine  cannot  work  with  equal  efficiency 
under  different  falls  unless  the  speed  is  varied  in  proportion 
to  the  square  root  of  the  fall.  Under  practical  working  con- 
ditions, however,  the  speed  is  nearly  always  required  to  main- 
tain a  constant  value,  while  the  fall  is  liable  to  fluctuations. 
This  usually  prevents  the  continuous  realization  of  the  highest 
efficiencies  of  which  a  turbine  may  be  capable. 

*  Lowell  Hydraulic  Experiments. 


EXAMPLES.  223 


EXAMPLES. 

1.  Let  the  dimensions  of  a  reaction  turbine  be  as  follows:  A\  =28°, 
n  =3.375  ft.,  r2  =4.146  ft.,  F,  =6.537  sq.  ft.,  /2  =7.687  sq.  ft.  (these  data 
are  the  same  as  those  of  Ex.  1,  Art.  222).  Suppose  the  discharge  to  take 
place  through  a  diffuser  whose  outer 'radius  exceeds  that  of  the  wheel 
by  3.5  ft.,  and  whose  vertical  dimension  diverges  from  .93  ft.  to  1.68  ft. 
Neglecting  losses  of  energy  in  the  diffuser,  and  assuming  k  =  .25,  deter- 
mine the  best  velocity  and  highest  efficiency. 

[Notice  that  if  losses  of  head  in  the  diffuser  are  neglected,  k'  depends 
only  upon  the  ratio  of  V2  to  the  velocity  of  outflow  from  the  diffuser. 
Calling  the  latter  F/,  and  assuming  F2  to  be  exactly  radial  (^.2=90°), 
the  above  dimensions  give 

F/     4.146  X. 93 


F2     7.646  Xl.( 


=  .300; 


hence  k'  =  .09.]  Ans.  x  =  .557,  y  =  .868,  e  =  .86. 

2.  Solve  the  preceding  example  on  the  assumption  that  A/ =  .5. 

Ans.  a:  =  .548,  2/  =  .857,  6  =  .83. 

3.  The  folowing  data  apply  to  a  reaction  turbine  with  axial  flow, 
discharging  into  the  air:    A,  =14°  40',  a2  =  153°  50',  Fi//2  =  .54,  mean 
radius  r  =.5  ft.     Determine  best  velocity,  highest  efficiency,  and  proper, 
value  of  vane  angle  ai,  assuming  £  =  .25. 

4.  The  following  data  are  for  a  mixed-flow  turbine  with  suction-tube ; 
4,=15°,  a2  =  160°,  F://2  =  .75,  r,/r,  =  1.5,  n=2.5  ft.,  ft  =30  ft.,  quantity 
of  water  available  120  cu.  ft.  per  sec.     Assume  £  =  .25,  A/ =  .9.     Deter- 
mine best  speed,  highest  efficiency,  best  value  of  a\.    Also  estimate  the 
values  of  Fi,  /i,  /2,  and  the  power  developed. 


CHAPTER  XX. 


TURBINE  PUMPS. 

231.  Relation  of  Pumps  and  Motors. — Turbine  pumps  are 
closely  related  to  turbine  motors  of  the  reaction  type.     It  is, 
in  fact,  quite  possible  to  design  a  turbine  which  may  operate 
either  as  motor  or  pump.     The  fundamental  equations  in  the 
theory  of  turbine  pumps  are  identical  with  those  already  given 
in  the  theory  of  reaction  turbines, 

except  that  certain  changes  of  nota- 
tion are  found  advisable.  It  will 
be  instructive,  however,  to  develop 
the  theory  of  pumps  independently, 
introducing  it  by  illustrations  of  an 
elementary  character. 

The  most  common  forms  of  tur- 
bine pumps  are  known  as  centrifugal 
pumps,  because  of  the  fact  that  an 
important  factor  in  their  operation 
is  the  variation  of  pressure  due  to 
rotation  (Art.  30). 

232.  Column  of  Water  Sustained 
by    "  Centrifugal    Force."— Let     a 
paddle-wheel  rotate  uniformly  in  a 
closed  cylindrical  chamber  filled  with 
water,   the    axis   of   rotation   being 
vertical  (see  horizontal  and  vertical 
sections,  Fig.  111).     The  water  will 

tend   to   take  up  a  condition  of  uniform  rotation  with  the 
wheel.    The  whole  body  of  water  will  not  rotate  as  a  rigid 

224 


FIG.  111. 


CRUDE  FORM  OF  CENTRIFUGAL  PUMP. 


225 


body,  because  of  friction  against  the  surface  of  the  chamber; 
but  approximately  such  a  condition  of  rotation  will  be  reached. 
If  vertical  tubes  communicate  with  the  chamber  at  A  and  B, 
water  will  stand  in  them  at  unequal  heights.  The  difference  in 
level  of  the  tops  of  the  two  columns  may  be  computed  from 
the  results  of  Art.  30.  If  u\  and  u2  are  the  velocities  of  the 
points  A  and  B  of  the  rotating  body,  the  pressure  head  at  B 
exceeds  that  at  A  by  the  amount  (u22-ui2)/2g-,  which  is  there- 
fore also  the  vertical  distance  of  B'  above  A' . 

233.  Crude   Form  of   Centrifugal  Pump. — Consider  next  a 
paddle-wheel  rotating  in  a  cylindrical  chamber  X  (Fig.  112), 


3D 


FIG.  112. 

surrounded  by  a  second  chamber,  Y,  with  which  it  communi- 
cates by  means  of  orifices.    A  vertical  pipe  is  connected  with 


226  TURBINE  PUMPS. 

the  chamber  Y,  and  another  pipe  leads  from  a  point  near  the 
middle  of  the  chamber  X  to  a  reservoir  R.  If  the  paddle-wheel 
is  stationary,  water  fills  both  chambers  and  rises  in  the  vertical 
pipe  to  a  point  P,  level  with  the  surface  of  the  water  in  the 
reservoir.  If  the  wheel  rotates  with  angular  velocity  &>,  the 
pressure  in  the  chamber  X  will  vary  according  to  the  law  already 
given.  If  r  is  the  radius  of  the  inner  chamber  (practically  equal 
to  that  of  the  wheel),  the  pressure  at  any  point  of  the  cylindrical 
bounding  surface  of  the  chamber  X  will  exceed  that  at  a  point 
in  the  axis  of  rotation  at  the  same  level  by  the  equivalent  of  a 
water  column  of  height  r2a2/2g.  Pressure  is  communicated  to- 
the  water  in  the  outer  chamber  through  the  connecting  orifices, 
and  a  condition  of  equilibrium  will  be  reached  in  which  the 
pressure  within  the  chamber  Y  and  the  vertical  pipe  follows 
the  hydrostatic  law.  At  any  point  in  this  outer  chamber  the 
pressure  will  equal  that  at  points  on  the  same  level  at  the  outer 
boundary  of  the  inner  chamber.  When  equilibrium  exists  in 
the  outer  chamber  the  pressure  conditions  in  the  different  parts 
of  the  apparatus  are  thus  seen  to  be  the  following: 

(a)  Throughout  the  reservoir  R  and  the  connecting  pipe, 
and  at  all  points  of  the  chamber  X  which  lie  in  the  axis  of  rota- 
tion, the  pressure  increases  with  the  depth  below  the  surface 
of  the  reservoir  according  to  the  hydrostatic  law.  (6)  Through- 
out the  chamber  Y  and  the  connecting  pipe  the  pressure  varies 
according  to  the  hydrostatic  law;  but  the  pressure  at  any  point 
exceeds  that  at  points  on  the  same  level  in  the  reservoir  72  by 
the  equivalent  of  a  water  column  of  height  r2w2/2g,  or  u2/2g 
(u  being  the  linear  velocity  of  the  outer  ends  of  the  wheel- 
blades)  . 

Evidently,  therefore,  water  will  rise  in  the  pipe  to  a  point 
Q,  such  that  PQ  =  u2/2g. 

Let  the  open  end  of  the  pipe  be  at  a  height  h  above  P.  If 
the  velocity  of  rotation  of  the  wheel  be  such  that 


water  will  rise  to  the  open  end  of  the  pipe;  and  if  the  velocity 


TRANSFORMATIONS  OF  ENERGY.  227 

of  rotation  exceeds  this  value,  water  will  flow  out.  In  this  way 
a  continuous  stream  can  be  caused  to  flow  from  the  reservoir, 
discharging  at  the  open  end  of  the  pipe;  the  possible  height  of 
the  lift  being  proportional  to  the  square  of  the  wheel  velocity. 
We  thus  have  a  centrifugal  pump. 

It  is  not  necessary  that  the  pump  shall  be  lower  than  the 
surface  of  the  supply  reservoir;  but  if  it  is  higher,  some  means 
must  be  provided  for  filling  the  chambers  of  the  pump  and  the 
pipe  leading  from  the  reservoir,  and  to  prevent  the  water  from 
running  back  before  the  wheel  velocity  becomes  great  enough 
to  sustain  it.  This  may  be  accomplished  by  a  foot- valve 
placed  in  the  supply  pipe,  so  arranged  as  to  open  only  in  the 
direction  of  flow.  Neither  is  it  necessary  that  the  axis  of  rota- 
tion be  vertical. 

234.  Transformations  of  Energy. — In  such  an  apparatus  as 
that  shown  in  Fig.  112  energy  must  constantly  be  supplied  to 
the  wheel  in  order  to  maintain  the  uniform  rotational  velocity. 
This  energy  is  -given  up  to  the  water  by  the  action  of  the  wheel- 
blades.  The  particles  of  water  next  to  the  paddles,  which 
receive  energy  directly  from  the  wheel,  pass  it  on  to  the  neigh- 
boring particles,  and  through  these  it  is  transmitted  to  the 
particles  remote  from  the  moving  blades.  Since  there  will 
always  be  some  relative  motion  of  the  particles  of  water  among 
themselves,  some  energy  is  dissipated  into  heat  by  reason  of 
internal  friction  in  the  water.  Another  portion  of  energy  is 
dissipated  by  reason  of  the  friction  between  the  rotating  body 
of  water  in  the  chamber  X  and  the  walls  of  the  chamber.  Such 
losses  occur  even  if  no  flow  takes  place  through  the  wheel. 

If  the  wheel  velocity  is  such  that  flow  takes  place,  other 
losses  of  energy  occur.  There  is  evidently  a  frictional  loss  such 
as  always  accompanies  the  flow  of  water.  Also,  since  a  particle 
flowing  through  the  apparatus  suffers  sudden  changes  of  velocity, 
there  is  a  loss  due  to  this  cause.  Following  a  particle  from  the 
point  where  it  enters  the  wheel  chamber  to  a  point  in  the  dis- 
charge-pipe, it  will  be  seen  in  a  general  way  how  its  velocity 
varies.  In  moving  from  the  axis  of  rotation  to  the  circum- 


228 


TURBINE  PUMPS. 


ference  of  the  chamber  X,  its  velocity  will  be  changed  gradually 
from  a  small  value  to  a  value  nearly  equal  to  that  of  the  outer 
ends  of  the  paddles.  As  it  enters  one  of  the  orifices  leading 
into  the  outer  chamber  it  is  suddenly  deflected;  and  on  enter- 
ing this  chamber  its  velocity  is  nearly  destroyed.  These  losses 
of  energy  are  so  important  that  such  a  crude  pump  as  that 
represented  in  Fig.  112  would  have  a  very  low  efficiency. 

235.  Design   for  Efficient  Pump.— In  Fig.  113*  are  shown 
the  main  features  of  a  pump  designed  to  reduce,  as  far  as  prac- 


FIG.  113. 

ticable,  the  losses  of  energy  by  dissipation  into  heat.  The  axis 
of  rotation  is  horizon tal,t  (A)  being  a  section  by  a  plane  con- 
taining this  axis,  and  (B)  a  section  by  a  plane  perpendicular 
to  the  axis.  The  rotating  wheel  is  formed  with  continuous 
passages,  bounded  by  curved  surfaces  in  such  a  way  as  to  cause 
a  gradual  deflection  of  the  water  instead  of  a  sudden  deflection. 
The  vanes  which  separate  the  passages  are  curved  "back- 
ward," i.e.,  in  such  a  way  that  the  relative  velocity  of  a  par- 
ticle of  water  when  it  reaches  the  outer  circumference  of  the 

*  This  figure  shows  the  essential  hydraulic  features  of  the  Risdon-Sulzer 
pump. 

t  The  direction  of  the  axis  is  not,  however,  essential. 


RELATION   BETWEEN   LIFT  AND  WHEEL  SPEED.        229 

wheel  is  directed  as  nearly  as  possible  opposite  to  the  velocity 
of  the  perimeter  of  the  wheel  at  that  point;  the  object  being 
to  make  the  absolute  velocity  of  outflow  as  small  as  possible. 

If  it  were  possible  to  save  all  the  kinetic  energy  possessed 
by  the  water  as  it  leaves  the  wheel,  the  direction  of  the  absolute 
velocity  of  outflow  would  be  immaterial,  so  far  as  efficient 
working  is  concerned;  it  might  be  advantageous  to  make  the 
angle  a2  as  small  as  90°,  or  even  smaller,  since  by  this  means 
the  rate  at  which  the  wheel  would  give  energy  to  the  water 
(when  running  at  a  given  speed)  would  be  increased.  The 
reason  for  avoiding  a  high  velocity  of  outflow  from  the  wheel 
is  that  an  important  fraction  of  the  kinetic  energy  due  to  this 
velocity  is  necessarily  lost. 

The  absolute  velocity  of  outflow  must,  however,  have  a 
forward  tangential  component,  since  the  head-equivalent  of  the 
energy  imparted  by  the  wheel  to  the  water  is  proportioned  to 
this  component  and  to  the  speed  of  rotation.*  It  is  important 
to  conduct  the  water  from  the  wheel  to  the  receiving  chamber 
in  such  a  manner  as  to  reduce  the  velocity  of  flow  as  gradually 
as  possible.  This  is  the  aim  of  the  design  shown  in  Fig.  113. 
The  water  passes  from  the  wheel  into  passages  conforming  to 
the  direction  of  its  absolute  velocity  just  before  leaving  the 
wheel,  and  these  passages  enlarge  very  gradually  to  their  place 
of  discharge  into  the  receiving  chamber. 

236.  Relation  between  Lift  and  Wheel  Speed.  —  It  was 
shown  above  (Art.  233)  that  the  height  of  the  column  of  water 
that  can  be  sustained  by  the  rotation  of  the  wheel  while  no 
flow  takes  place  is  u22/2g,  where  u2  is  the  velocity  of  the  outer 
ends  of  the  wheel-blades.  When  flow  occurs,  however,  this 
i elation  no  longer  holds.  Energy  is  continually  given  up  by 
the  wheel  to  the  water,  and  this  results  in  a  gain  of  effective 
head  which  may  take  the  form  of  an  increased  lift.  If  hm  is 
the  head  equivalent  to  the  energy  imparted  to  the  water  by  the 
wheel  (i.e.,  the  energy  given  up  by  the  wheel  per  pound  of 
water  discharged);  hf  the  total  loss  of  head  by  reason  of  dis- 
sipation of  energy  between  supply  reservoir  and  discharge 
*  This  follows  from  formula  (IV),  as  will  be  shown  below. 


230 


TURBINE  PUMPS. 


reservoir;  HI  and  H2  the  values  of  the  effective  head  at  sur- 
faces of  supply  reservoir  and  discharge  reservoir  respectively; 
the  general  equation  of  energy  gives  (Art.  96) 

H2-Hi=h'"-h'. 

The  lift  H2-Hi  is  thus  equal  to  the  head-equivalent  of  the 
energy  supplied  by  the  wheel  diminished  by  the  frictional  losses 
of  head.  If  a  wheel  is  so  designed  and  operated  as  to  impart 
energy  to  the  stream  efficiently  (i.e.,  with  small  loss  by  dissipa- 
tion), the  lift  maintained  while  flow  occurs  may  be  materially 
greater  than  that  due  to  rotation  when  there  is  no  flow.  This 
is  borne  out  by  experience,  some  of  the  best  pumps  being  found 
to  give  a  lift  greater  than  u22/2g. 

237.  Notation  for  Mathematical  Theory.— In  the  following 
mathematical  theory  the  notation  used  in  the  theory  of  the 
reaction  turbine  will  be  employed,  with  the  following  modifica- 
tions. 

We  now  have  to  deal  with  a  lift  instead  of  a  fall,  and  with 
energy  given  up  by  wheel  instead  of  energy  received  by  wheel  from 
water.  Hence  we  shall  put 

L  =  energy  given  up  by  wheel  per  second; 
h  =  total  lift  from  surface  of  supply  reservoir  to  surface 
of  discharge  reservoir. 

No  change  is  made  in  the  notation  for  absolute  and  relative 
velocities  and  their  direction  angles.     It  is, 
however,  found  convenient  to  express  the 
equations  in  terms  of  u2  and  v2,  instead 
of  ui  and  Vi  as  in  the  former  theory. 

The  discussion  will  refer  to  the  case  in 
which  the  admission  and  discharge  occur 
substantially  as  in  Fig.  113.  The  absolute 
velocity  V\  thus  means  the  velocity  of  flow 
in  the  suction-pipe  close  to  the  wheel,  and 
FI  means  the  cross-section  at  this  pointe 
Since  V\  is  parallel  to  the  axis  of  ro-  FIQ 

tation,    ^i=90°    and    cos   Ai=0.      This 
greatly  simplifies  the  main  fundamental  equations. 


..1 


FUNDAMENTAL  EQUATIONS.  ^     231 

The  direction  of  flow  in  the  passages  leading  from  the  wheel 
to  the  receiving  chamber  depends  upon  the  form  of  the  pas- 
sages. For  the  purpose  of  the  mathematical  theory  it  is 
necessary  to  idealize  the  conditions  of  flow  within  these  pas- 
sages by  assuming  that  the  direction  angle  of  the  velocity  of  a 
particle  immediately  after  leaving  the  wheel  has  the  same  value 
for  all  particles.  This  angle,  measured  as  usual  from  the  direc- 
tion of  u2,  will  be  called  A2 '.  It  must  be  remembered  that  A2 
is  the  direction  angle  of  V2,  which  is  the  absolute  velocity  of  a 
particle  just  before  leaving  the  wheel.  For  a  given  pump  A2 
and  A2  can  be  equal  only  for  some  one  ratio  of  wheel  velocity 
to  lift  velocity.  It  is  desirable  that  they  should  be  equal  when 
this  ratio  has  the  value  under  which  the  pump  is  designed  to 
operate,  since  if  they  are  unequal  a  sudden  change  of  velocity 
occurs  which  results  in  loss  of  energy. 

238.  Fundamental  Equations. — The  energy  given  up  by  the 
wheel  to  the  water  is  expended  in  two  ways:  in  lifting  the 
water  a  height  h,  and  in  overcoming  the  wasteful  resistances 
represented  by  the  lost  head  h'.  Hence 

L  =  W(h+h') (1) 

The  value  of  L  may  also  be  expressed  by  a  formula  similar 
to  (IV)  of  Art.  179.  In  fact  the  reasoning  leading  to  that 
formula  (Art.  177)  is  strictly  applicable  to  the  case  in  which 
the  energy  given  to  the  wheel  by  the  water  is  negative.  But 
with  the  present  meaning  of  L  we  must  change  signs  and  write 

W 

L= — 

y 

Or,  since  now  «i  =  Vi  cos  AI  =  0, 

W 
we  have  L  =  —u2S2 (2) 

y 

But  s2  =  u2  +  v2  cos  a2 ; 


232  TURBINE  PUMPS. 

W 
hence  L=—  ^2(^2  +  ^2  cos  a2)  ......    (3) 

t7 

Equations  (1)  and  (3)  now  give 

g  (h  +hf)  =u2(u2~\-V2  cos  c^)  .....     (4) 

Equations  (3)  and  (4)  are  the  fundamental  equations  upon 
which  the  following  theory  is  based. 

Equation  (2)  shows  that  if  L  has  a  positive  value,  s2  must 
be  positive;  thus  justifying  the  statement  made  in  Art.  235 
that  the  water  must  leave  the  wheel  with  a  velocity  having  a 
forward  tangential  component. 

239.  Loss  of  Head.  —  To  complete  the  theory  it  is  necessary 
to  estimate  the  value  of  h'.  Doubtless  the  most  important 
losses  of  head,  in  a  pump  working  at  its  best  speed  for  the  exist- 
ing lift,  are  of  the  kind  included  under  the  term  friction  losses. 
These  depend  upon  the  velocity  of  flow  and  the  character  of 
the  passages,  and  may  be  expressed  by  means  of  a  single  term 


in  accordance  with  the  usual  assumption  regarding  such  losses. 
If  there  is  a  sudden  change  in  the  direction  of  the  velocity 
of  particles  leaving  the  wheel,  this  causes  a  loss  which  cannot 
be  expressed  by  a  term  of  the  above  form.  The  passages  into 
which  discharge  occurs  should,  if  possible,  be  so  formed  that 
such  sudden  deflection  of  the  water  does  not  take  place.  This 
condition  cannot  be  satisfied  except  for  a  particular  relation 
between  speed  and  lift,  which  should  be  that  for  which  the 
pump  is  designed  to  work.  In  discussing  the  question  of  design 
for  highest  efficiency,  it  will  be  instructive  to  give  first  a  solu- 
tion of  the  problem  assuming  that  no  sudden  deflection  of  the 
outflowing  stream  occurs.*  Although  the  resulting  equations 
will  not  show  the  working  of  an  actual  pump  for  different  rela- 

*  This  corresponds  to  the  solution  given  in  Art.  218  for  turbine  motors. 


DESIGN   FOR  HIGHEST  EFFICIENCY.  233 

tions  of  speed  to  lift,  they  should  agree  well  with  actual  con- 
ditions for  the  case  which  is  to  govern  the  design.  The  question 
of  efficiency  curves  and  discharge  curves  for  a  given  pump  will 
be  considered  afterward. 

240.  Design  for  Highest  Efficiency.  —  Substituting  in  equa- 
tion (4) 


we  obtain 

2u22  +  2u2v2  cos  a2  -  kv22  =  2gh,     ....     (5) 

from  which  v2  and  the  rate  of  discharge  can  be  computed  for 
any  wheel  velocity. 

The  value  of  the  efficiency  may  be  expressed  as  follows: 
Since  the  energy  utilized  per  second  is  Wh,  while  the  energy 
supplied  by  the  wheel  to  the  water  is  W(h+h'),  the  hydraulic 
efficiency  is 

Wh  h 

~W(h+h')~h+h'>     •••.•••    (°) 

or  from  (4)  ,  e  =  —,  -  -^  -  ,  .  (7) 

^2(^2  +  ^2  cos  $2) 

The  efficiency  for  any  wheel  speed  is  found  by  using  in  (7)  the 
value  of  v2  in  terms  of  u2  determined  from  equation  (5). 

The  mathematical  condition  for  maximum  efficiency  is 
de/du2  =  Q.  The  solution  may  be  carried  out  by  the  method 
employed  in  Art.  220.  In  the  present  problem,  however,  the 
following  method  is  shorter. 

v  2 

Since  hf  =  k-^-  ,  the  value  of  e  in  (6)  may  be  written 
*9 

2gh 


This  equation  shows  that,  for  any  given  value  of  h,  e  decreases 
continually  as  v2  increases,  having  its  greatest  value  when  v2  =  0. 
The  maximum  value  of  e  is 

em  =  l,     ........     (9) 


234  TURBINE  PUMPS. 

while  the  corresponding  value  of  u2,  found  by  putting  v2=Q 
in  (5),  is 

u2=Vgh,      .......    (10) 

The  angle  of  outflow  of  the  water  from  the  wheel  is 

A2=0,    .    .    .    .....    (11) 

as  may  be  seen  from  the  relation  between  the  three  vectors 
representing  u2,  v2,  V2.    Thus,  since  always 


],        ......      (12) 

the  condition  v2=Q  gives 

[V2]=[u2]  ........    (13) 

The  meaning  of  these  results  is  more  easily  seen  from  a 
graphical  representation. 

Let  m=V2gh,  x  =  u2/m,  y  =  v2/m.  Equations  (5)  and  (7) 
may  then  be  written 


(14) 


Taking  x  and  y  as  rectangular  coordinates,  equation  (14) 
represents  an  hyperbola  whose  center  is  at  the  origin.  A  par- 
ticular case  of  this  curve  is  represented  at  (A)  in  Fig.  115, 
which  shows  only  the  part  of  the  curve  for  which  y  is  positive. 
One  asymptote  of  the  hyperbola  is  shown  in  the  figure. 

Taking  values  of  e  as  ordinates  of  a  curve  of  which  the 
abscissas  are  the  corresponding  values  of  x,  the  result  is  shown 
at  (B),  Fig.  115. 

For  a  constant  value  of  the  lift  the  rate  of  discharge  varies 
as  y,  and  the  speed  as  x.  Hence  curves  (A)  and  (B)  show  how 


DESIGN  FOR  HIGHEST  EFFICIENCY. 


235 


the  rate  of  discharge  and  the  efficiency  vary  with  the  speed 
when  the  lift  remains  constant. 

It  must  be  remembered  that  these  results  do  not  apply  to 
a  given  pump  working  at  different  speeds,  but  show  the  com- 
parative results  of  an  assumed  series  of  pumps  running  at  differ- 
ent speeds,  assuming  that  in  every  case  the  absolute  velocity 


1.00 


Values  of  x. 


1.0 
FIG.  115. 


2.0 


of  outflow  receives  no  sudden  change,  but  that  the  flow  from 
the  wheel  to  the  receiving  chamber  occurs  with  equal  efficiency 
for  all. 

S.trictly  interpreted,  the  conclusion  is  that  the  case  of  maxi- 
mum efficiency  is  that  in  which  the  angle  A 2  is  zero,  the  cor- 
responding wheel  velocity  being  Vgh  or  .707m,  and  the  rate 
of  discharge  zero.  As  the  ratio  u2/m  or  x  increases,  the  effi- 
ciency decreases  and  the  rate  of  discharge  increases,  the  angle 
A2  also  increasing. 

The  practical  conclusion  is  that  the  case  of  maximum  effi- 
ciency is  an  ideal  limit  toward  which  the  design  should  approxi- 
mate, but  which  it  can  never  reach,  and  that  the  speed  of  work- 
ing should  be  such  as  to  give  a  relatively  small  value  of  the  dis- 
charge angle  A  2.  But  in  order  to  produce  a  given  rate  of 
discharge  with  as  small  a  pump  as  practicable,  it  may  be  advis- 
able to  let  the  working  speed  increase  materially  above  the 
value  corresponding  to  the  ideal  case  of  maximum  efficiency. 


236  TURBINE  PUMPS. 

241.  Losses  of  Head  in  a  Given  Pump. — In  an  actual  pump 
wor Icing  under  different  relations  of  speed  to  lift,  the  entire 
loss  of  head  cannot  be  expressed  by  a  term  of  the  form  assumed 
above.  In  addition  to  the  " friction"  losses  there  are  losses 
due  to  sudden  deflections  of  the  stream  as  it  enters  and  as  it 
leaves  the  wheel,  such  deflections  depending  not  upon  the 
velocity  of  flow  alone,  .but  also  upon  the  speed  of  rotation. 
Since  the  water  enters  the  wheel  near  the  axis  of  rotation 
where  the  wheel  speed  is  relatively  small,  the  deflection  of  the 
stream  due  to  lack  of  exact  adjustment  of  vanes  to  stream 
velocity  at  this  point  is  probably  of  small  importance.  Sudden 
deflection  at  the  point  of  outflow  from  the  wheel  would  appear, 
however,  to  be  an  important  cause  of  loss  of  head.  While  no 
exact  evaluation  of  this  loss  can  be  made  on  theoretical  grounds 
alone,  the  following  considerations  lead  to  an  expression  which 
probably  represents  actual  conditions  at  least  approximately 
for  a  pump  discharging  in  the  way  shown  in  Fig.  113. 

It  may  be  noticed  first  that  in  the  limiting  case  of  no  dis- 
charge (02  =  0)  we  ought  to  have  h'  =  u22/2g.  For  in  this  case 
the  gain  of  effective  head  due  to  the  action  of  the  wheel  (which 
is  always  the  value  of  L/W  given  by  (3))  reduces  to  u22/g, 
while  the  actual  lift  has  the  value  u22/2g  by  Art.  233,  so  that 
the  lost  head  is  the  difference  between  these  values.  That 
u22/2g  is  a  reasonable  value  for  hf  in  this  limiting  case  is  seen 
also  by  considering  what  occurs  when  there  is  a  very  slight 
discharge.  A  particle  just  about  to  leave  the  wheel  has 
absolute  velocity  u2,  while  just  outside  the  wheel  it  enters 
a  body  of  water  at  rest,*  so  that  its  kinetic  energy  is 
wholly  lost. 

*  Of  course  this  is  not  an  accurate  statement  of  actual  conditions,  since 
the  water  in  the  passages  just  outside  the  wheel  would  be  set  in  motion  by 
the  friction  of  the  rotating  body.  With  such  a  construction  as  that  shown 
in  Fig.  113,  however,  this  water  is  prevented  from  taking  up  the  motion  of 
the  wheel  periphery,  and  the  statement  that  the  velocity  of  the  particle 
leaving  the  wheel  is  wholly  destroyed  is  practically  true.  The  assumption 
above  made  as  to  the  loss  of  head  is  in  accordance  with  the  usual  rule  for 
estimating  loss  of  head  due  to  a  stream  entering  a  body  of  water  at  rest. 
(Art.  82.) 


EQUATIONS  FOR  RATE  OF  DISCHARGE  AND  EFFICIENCY.    237 


Considering  now  the  general  case  in  which  u2  and  v2  have 
any  values,  we  will  make  the  ideal  assumption  referred  to  in 
Art.  237,  that  the  direction  angle  of  the  absolute 
velocity  of  a  particle  of  water  just  after  leaving 
the  rotating  wheel  has  the  same  value  for  all 
particles,  this  value  being  fixed  by  the  form  of 
the  passages  leading  to  the  receiving  chamber. 
In  other  'words,  it  will  be  assumed  that  the 
absolute  velocity  changes  from  a  value  V2  with 
direction  angle  A 2  to  a  value  V2  with  direction 
angle  A2)  and  that  the  latter  angle  has  a  fixed 
value,  while  the  former  varies  with  u2  and  v2.  In 
Fig.  116,  AB,  BC  and  AC  represent  u2,  v2  and  V2 
respectively;  while  EAC'  is  the  fixed  angle  A2, 
and  V2  is  represented  by  some  vector  AC' '.  The 
absolute  velocity  thus  changes  from  AC  to  AC' 
as  the  particle  leaves  the  wheel. 

It  will  be  assumed  that  the  loss  of  head  due  to  this  sudden  change 
is  (CC')2/2g. 

The  value  of  CCr  is  to  be  expressed  in  terms  of  u2  and  v2f 
Since  with  proper  construction  (the  thickness  of  the  wheel- 
passages  being  equal  to  that  of  the  passages  into  which  the 
discharge  occurs)  the  component  of  V2  and  that  of  V2  per- 
pendicular to  u2  are  equal,  CC'  is  parallel  to  AB,  and  its  value 
is  easily  found  to  be 


sin(a2-A2') 


sin  A  2 


=  u2-k'v2,  .     .     .     (16) 


if  k'  is  written  for  the  constant 


sin  («2— -A 2') 


The  total  loss  of  head  may  now  be  expressed  by  the  equation 


2g 


(17) 


242.  Equations   for   Rate   of    Discharge    and   Efficiency.— 
Substituting  the  above  value  of  h'  in  equation  (4),  the  latter 


238  TURBINE   PUMPS. 

takes  the  form 

.    .     (18) 


The  value  of  the  hydraulic  efficiency  in  terms  of  u2  and  v2, 
as  given  by  equation  (7),  is  not  changed,  but  in  applying  it  v2 
must  have  values  satisfying  (18).  The  equation  is 


cos  a2)' 


(19) 


Introducing  ratios  of  velocities,  x  and  y  having  meanings  as 
in  Art.  240,  the  above  equations  become 

y2  =  l;    .    .    .     (20) 


243.  Condition  of  Maximum  Efficiency.  —  The  values  of  x 
and  y  for  which  e  is  a  maximum  may  be  found  by  applying  the 
condition  de/dx  =  0.  Writing  (21)  in  the  form 


cosa2 

4O 

and  differentiating, 

1    de     ,n  v  .  d?/     . 

-•^^-  =  (2x  +  ycosa2)+xcosa2--j-=Q. 


Differentiating  (20), 

[x  +  (kf  +cos  a2)y]  +[(k'  +cos  a2)x  - 


Equating  values  of  dy/dx  given  by  these  two  equations,  and 
reducing, 


=o.    .    .    .     (22) 


GRAPHICAL  REPRESENTATION   FOR  CONSTANT  LIFT.     239 

This  equation  determines  two  values  of  y/x,  one  of  which  gives 
a  real  solution  of  the  problem.  Calling  this  value  a,  and  putting 
y=ax  in  equation  (20),  the  result  is 


]x2  =  l,  .     .     .     (23) 
which  gives  the  value  of  x  for  maximum  efficiency. 

244.  Graphical  Representation  for  Case  of  Constant  Lift.— 

When  the  lift  remains  constant,  x  varies  directly  as  the  wheel 
speed,  and  y  as  the  rate  of  discharge.  Hence  the  curve  repre- 
sented by  (20),  when  x  and  y  are  made  rectangular  coordinates, 
shows  how  the  rate  of  discharge  varies  when  the  wheel  is  run 
at  different  speeds  under  a  constant  lift. 

The  relation  between  efficiency  and  wheel  speed  under  the 
same  conditions  may  be  represented  by  a  curve  of  which  the 
ordinate  and  abscissa  of  any  point  are  corresponding  values  of 
e  and  x.  The  value  of  e  for  any  value  of  x  is  to  be  computed 
from  (20)  and  (21). 

A  third  curve  of  importance  is  that  showing  the  relation 
between  wheel  velocity  and  power.  This  relation  is  expressed 
by  equation  (3),  or  more  conveniently  by  the  equivalent  equa- 
tion 

Wh 


Since  W  varies  with  varying  speed,  we  substitute  for  it  its 
value, 

W  =  wq  =  wf2v2  =  wf2my, 

with  the  result 

T     w}2myh    wj2m3  y     wf2ms, 

L  = =  -^ -  =  -75 — I       .    .     .     (24) 

e  2g      e        2g 

The  variable  factor  in  this  value  of  L  is  y/e,  which  has  been 
represented  by  1.  If  I  be  made  ordinate  of  a  point  whose  ab- 
scissa is  x,  the  locus  of  this  point  as  x  varies  is  a  curve  showing 
the  way  in  which  the  power  varies  with  the  wheel  velocity. 


240  TURBINE  PUMPS. 

The  equations  for  the  case  of  constant  lift  will  for  convenience 
be  summarized  below.  With  them  will  be  included  (22),  which 
applies  to  the  case  of  maximum  efficiency,  and  which  is  seen  to 
represent  two  straight  lines,  one  of  which  intersects  the  general 
x-y  curve  in  the  point  whose  abscissa  is  the  value  of  x  for 
maximum  efficiency.* 


l.    .    .    f     (A) 


2x(x+ycosa2)'    •••••- 

a2x2  =  <).    .    ^     .     (Q 

(D) 


_     wf2m?  y  n     ,  N  1 

L=          '       --2*2/(*+2/cosa2)=--Z.     .    (E) 


Special  case.  —  For  a  given  wheel  the  angles  a2  and  A2  are 
known,  and  k'  can  be  computed  from  them.  The  friction 
factor  k  cannot,  however,  be  known  apart  from  experiment. 
Consider  a  pump  for  which 


Equations  (A),  (B),  and  (C)  become 

l',       .    .    .     (25) 


e==2x(x-M6y)'     ......     (26) 

7  ^5 
2f>  -2.31*2,  +  ^3-7^=0  .....    (27) 

*  It  will  be  noticed  that  there  is  a  correspondence  between  these  equa- 
tions and  those  given  in  Art.  223  for  the  reaction  turbine. 


GRAPHICAL  REPRESENTATION  FOR   CONSTANT  LIFT.     241 

It  appears  that  if  k  is  small  in  comparison  with  13.70,  it 
has  little  influence  on  the  form  of  the  x-y  curve.  Fig.  117 
shows  the  curves  obtained  by  assuming*  &=0.  This  reduces 
(25)  and  (27)  to  the  forms 


(28) 


(29) 


while  (26)  is  unchanged  since  it  does  not  involve  k  explicitly. 


l.O 


IE), 


(A) 


(B) 


Values  of  x 


1.0 
FIG.  117. 


2.0 


Equation  (28)  represents  an  hyperbola  whose  center  is  at 
the  origin  of  coordinates.  The  curve  cuts  the  z-axis  in  the 
point  x  =  l,  and  the  asymptotes  are  the  lines 


=  .547z, 


The  coordinates  of  the  vertex  are  £  =  .795,  t/  =  .148. 

A  remarkable  feature  of  the  curve  is  that  for  a  certain  range 
of  values  of  x  there  are  double  values  of  y\  i.e.,  to  each  value 


*  It  is  evident  that  this  assumption  makes  the  maximum  efficiency  1, 
since  h'  will  reduce  to  0  when  A2=A2', 


242 


TURBINE  PUMPS. 


.2 


of  the  wheel  velocity  there  correspond  two  values  of  the  rate 
of  discharge.    Although  this  may  at  first  sight  seem  anomalous-, 

it  is  in  harmony  with  the  general 
explanation  given  in  Art.  236.  If 
.70  the  height  of  the  column  sustained 
by  rotation  without  flow  is  u22/2g 
(in  accordance  with  the  theory  of 
Art.  233),  2/  =  0  should  give '  x  =  l. 

But  when  flow  occurs,  the  lift  may 
60 

be  greater   than    u22/2g    because  of 

the  energy  imparted  by  the  rotating 
e  wheel  to  the  water  flowing  through 
the  wheel  passages;  that  is,  values  of 
.50  x  less  than  1  would  give  positive 
values  of  y.  The  soundness  of  this 
reasoning  is  corroborated  by  the  re- 
sults of  experiments.  Fig.  118 
shows  the  x-y  curve  and  the  effi- 
ciency curve  *  obtained  from  experiments  on  a  pump  similar 
in  design  to  the  one  considered  in  the  foregoing  theory. 

The  curves  in  Fig.  117  are  marked  with  the  same  letters  as 
the  corresponding  general  equations. 
One  solution  of  equation  (29)  is 


.1 


x  l.O 

FIG.  118. 


which  represents  a  straight  line  intersecting  the  x-y  hyperbola 
in  the  point  whose  coordinates  are  x  =  .808,  y  =  .218,  the  values 
corresponding  to  maximum  efficiency.  The  value  of  the  maxi- 
mum efficiency  is  1,  as  it  must  be  if  &  =  0. 

245.  Effect  of  Friction  Loss  on  Form  of  Curves. — In  order 
to  show  how  the  form  of  the  curves  and  the  solution  for  maxi- 

*  It  should  be  said  that  the  efficiencies  represented  in  the  experimental 
curve  are  gross  efficiencies  computed  from  the  actual  work  done  in  driving 
the  pump,  while  the  efficiencies  given  by  equation  (B)  and  represented  by 
the  corresponding  curve  in  Fig.  117  are  of  course  hydraulic  efficiencies.  The 
experimental  curve  is  based  upon  data  supplied  by  Mr.  C.  H.  Stoddard,  Chief 
Mechanical  Engineer  of  the  Risdon  Iron  Works,  San  Francisco. 


REMARK  ON    FRICTION   LOSSES. 


243 


mum  efficiency  are  affected  by  the  friction  factor  k,  which  has 
been  assumed  zero  in  the  foregoing  example,  the  curves  for  k  =  5 
(all  other  data  remaining  unchanged)  are  shown  in  Fig.  119. 
Equations  (25)  and  (27)  now  become 


1.0 


CE) 


Values  01  x 


1.0 
FIG.  119. 


2.0 


(30) 
(31) 


The    last    equation   gives   for   maximum    efficiency   ?/  =  .189x, 
which  with  (30)  and  (26)  gives  z  =  .843,  ?/  =  .160,  e  =  M. 

246.  Remark  on  Friction  Losses.  —  In  the  above  theory  h 
was  defined  as  the  total  lift  from  supply  reservoir  to  discharge 
reservoir,  and  h'  as  the  total  head  lost  between  supply  and 
discharge  reservoirs.  The  reasoning  is  not  changed,  however, 
if  h  =  H2-Hi,  where  Hl  and  H2  are  the  values  of  the  effective 
head  at  two  sections  A  and  B  taken  anywhere  on  the  suction  and 
discharge  sides  of  the  pump  respectively,  and  h'  is  defined  as  the 
loss  of  head  between  these  sections.*  The  value  of  v2  for  any 


*That  is,  H^z^p./w  +  vS/Zg,  H2  =  z2  +  p2/w  +  v22/2g,  If  H,  and  H2 
are  to  be  determined  by  experiment,  it  is  advantageous  to  have  the  cross- 
sections  at  A  and  B  equal,  so  that  the  velocities  need  not  be  considered  in 
determining  H,  -  H^ 


244  TURBINE  PUMPS. 

given  value  of  u2  will  not  be  changed  by  changing  the  positions 
of  the  sections  A  and  B,  since  the  term  k(v22/2g)  expressing 
friction  loss  will  increase  or  decrease  exactly  as  H2-Hi  de- 
creases or  increases.  The  values  of  x  and  y  will,  however, 
change,  since  these  are  ratios  of  u2  and  v2  to  h.  The  compu- 
ted efficiencies  will  also  depend  upon  the  position  of  A  and  B, 
since  the  entire  loss  of  head  between  these  sections  is  charged 
against  the  pump  when  the  efficiency  is  computed  from  equa- 
tion (B). 

EXAMPLES. 

1.  The  following  are  dimensions  of  a  centrifugal  pump:   r2=6.5"  = 
.5417';    a2  =  150°;   /2  =  .108  sq.  ft.;    A2'  =  lo°.     Determine  x  and  y  for 
maximum  efficiency,  neglecting  friction  losses. 

Ans.  k'  =2.732,  z  =  .857,  y  =  .3U,  e  =  l. 

2.  Solve  with  same  data  except  assume  fc=6. 

Ans.  z  =  .903,  ?/  =  .168,  e  =  .73. 

3.  With  data  as  in  Ex.  2,  draw  curves  of  discharge,  efficiency  and 
power  under  constant  lift.  Ans.  x2  +  3.732xy  -  13.46y2  =  1. 

4.  With  same  data,  if  the  lift  is  150  ft.,  what  number  of  R.P.M. 
should  give  highest  efficiency,  and  what  would  be  the  corresponding  rate 
of  discharge?     If  the  speed  were  increased  so  as  to  double  the  rate  of 
discharge,  what  would  be  the  efficiency? 

Ans.  For  maximum  efficiency   .V  =  1565  R.P.M. ,  q  =  1.785  cu. 
ft.  per  sec. 

N  =  1870  R.P.M.  gives  g  =3.570  cu.  ft.  per  sec.,  e  =  .59. 

5.  Solve  examples  1  and  2  assuming  ^42'=30°. 

6.  The  following  are  dimensions  of  a  pump:   r2=8";   a2  =  156°;  /2  = 
5.25  sq.  in.;  A2'=25°.     (a)  Determine  x  and  y  for  maximum  efficiency  if 
k=4.     (b)  Determine  best  speed  of  rotation  and  rate  of  discharge  when 
the  lift  is  85  ft. 

247.  Curves  for  Rate  of  Discharge,  Efficiency  and  Power 
when  the  Wheel  Velocity  Remains  Constant. —  An  important 
practical  question  relates  to  the  effect  of  variations  of  the  lift 
when  the  pump  runs  at  constant  speed.  The  equations  apply- 
ing to  such  a  case  may  be  derived  from  those  already  given. 
It  will  be  convenient  to  introduce  the  variable  m/u2  =  I/x  =  x' 
instead  of  x,  so  that  x'  is  proportional  to  V2gh  or  m.  Further, 
the  rate  of  discharge  is  not  proportional  simply  to  y  when  the 


CURVES  FOR  RATE  OF  DISCHARGE.  245 

lift  varies,  since  m  in  equation  (D)   is  not  constant.     Since 
m  =  u2/x,  (D)  may  be  written 


We  therefore  introduce  y'  =  y/x  as  a  new  variable  to  represent 
the  rate  of  discharge.  The  five  main  equations,  expressed  in 
terms  of  x'  and  y',  become 


=  l>,  .    .     .     (A') 

x'2 


=0;.    ...     ((7) 
.    .    (D') 


In  (E'),  V  is  written  for  the  variable  factor  in  the  value  of  L; 
it  is  equal  to  l/x3. 

If  curves  are  constructed  with  xf  as  abscissa  and  yf,  e,  I', 
respectively,  as  ordinates,  they  will  show  the  variation  of  rate 
of  discharge,  efficiency  and  power  with  the  lift  when  the  speed 
remains  constant. 

Special  case.  —  For  the  particular  data  given  above  equa- 
tions (A')  and  (B')  become 

0/2  +  (13.70  +%'2-5.672/'  =  l,     ....     (32) 

x'2 


2(1 -.8662/')' 


(33) 


Assuming  as  before  &  =  0,  the  curves  are  shown  at  (A'),  (B'), 
(E'),  Fig.  120.    The  equation 


246 


TURBINE    PUMPS. 


represents  an  ellipse  whose  principal  axes  are  parallel  to  the 
axes  of  coordinates,  and  whose  center  is  at  the  point  x'  =  0, 


Of  special  interest  is  the  form  of  curve  (E').    This  indicates 
that  if  the  speed  is  constant,  an  increase  in  the  lift  causes  a 


decrease  in  the  power.*  This  explains  what  has  sometimes 
been  regarded  as  an  anomaly  in  the  practical  working  of  cen- 
trifugal pumps.  At  first  sight  it  might  be  inferred  that  an 
increased  lift  would  increase  the  load  upon  the  motor  which 
drives  the  pump,  but  the  opposite  effect  is  often  observed. 
This  is  in  accordance  with  curve  (E'),  and  is  easily  understood 
when  it  is  considered  that  the  power  is  proportional  directly 
to  the  rate  of  discharge  and  to  the  lift,  and  inversely  to  the 
efficiency.  An  increase  in  the  lift  (the  speed  remaining  con- 
stant) causes  a  decrease  in  the  rate  of  discharge,  and  probably 
also  an  increase  in  the  efficiency,  so  that  the  net  result  may  be 
to  decrease  the  power. 

EXAMPLE. 

With  data  as  in  Ex.  2,  Art.  246,  draw  curves  of  discharge,  efficiency 
and  power  for  constant  speed. 

*  Except  for  very  small  values  of  the  rate  of  discharge.  , 


COMPOUNDING. 


247 


248.  Compounding. — A  compound  pump  consists  of  two  or 
more  pumps  arranged  in  series,  i.e.,  so  that  the  discharge  of 
one  is  the  supply  of  the  next.  Since  the  lift  due  to  a  wheel 
does  not  depend  upon  the  actual  value  of  the  pressure  upon  it, 
the  total  lift  due  to  such  a  series  is  the  sum  of  the  lifts  which 
the  several  wheels  would  produce  acting  separately.  For  high 
lifts  compounding  is  common,  extremely  high  wheel  velocities 
being  thus  avoided.  The  different  wheels  are  commonly  made 
alike  in  dimensions  and  mounted  on  the  same  shaft,  so  that 
they  rotate  together  and  produce  equal  lifts.  See  Fig.  121. 


FIG.  121.     Worthington  three-stage  turbine  pump. 

249.  Balancing.— Since  the  impeller  must  have  clearance 
space  in  order  to  rotate  freely,  it  will  be  surrounded  by  water 
under  pressure.  Since  friction  will  prevent  this  water  from 
acquiring  the  full  rotational  velocity  of  the  wheel,  the  intensity 
of  pressure  throughout  the  clearance  space  may  approach  the 
value  existing  at  the  periphery  of  the  wheel.  The  intensity  of 
pressure  in  the  supply  pipe  being  much  less  than  this,  the  central 
portion  of  the  wheel  will  experience  a  much  greater  total  pressure 
from  the  back  than  from  the  front,  thus  subjecting  the  shaft  to 
a  thrust  which  must  be  provided  for  in  the  design. 


248 


TURBINE   PUMPS. 


In  order  to  balance  this  thrust  a  common  practice  is  to 
mount  upon  the  same  shaft  two  wheels  alike  in  all  respects 
except  that  they  are  right-  and  left-handed.  Fig.  122  shows 


FIG.  122.     Risdon-Sulzer  four-stage  pump  with  balanced  impellers. 

such  a  construction  with  two  pairs  of  balanced  runners,  con- 
nected in  series  so  as  to  form  a  four-stage  pump. 

250.  Actual  Lifts  and  Efficiencies. — It  was  formerly  sup- 
posed that  turbine  pumps  could  be  efficiently  used  only  with 
small  lifts  and  large  discharges.  Experiment  has  shown,  how- 
ever, that  with  proper  design  good  efficiencies  may  be  realized 
with  very  considerable  lifts.  Probably  hydraulic  efficiencies  as 
high  as  80  per  cent  can  be  obtained  under  a  lift  of  more  than 
100  ft.,  with  a  single  impeller  wheel.  By  compounding,  a  lift 
of  several  hundred  feet  is  entirely  practicable. 


APPENDIX  A. 
STEADY  FLOW  OF  A  GAS. 

A  1.  Effect  of  Compressibility  on  Theory  of  Steady  Flow.— 

When  the  fluid  is  compressible  the  theory  of  steady  flow  requires 
important  modification  in  two  particulars.  First,  it  is  no 
longer  necessary  that  equal  volumes  pass  different  sections  in 
a  given  interval  of  time.  Second,  as  the  density  of  any  definite 
portion  of  the  fluid  will  in  general  change  as  it  moves  along  the 
stream,  it  will  do  positive  or  negative  work  against  the  pres- 
sures acting  on  its  bounding  surface,  resulting  in  the  develop- 
ment of  mechanical  energy  if  the  body  expands,  and  the  absorp- 
tion of  mechanical  energy  if  it  contracts.*  Thus  both  the 
equation  of  continuity  and  the  general  equation  of  energy  will 
be  changed. 


Equation  of  Continuity.  —  If  the  flow  remains  steady, 
equal  masses  of  fluid  must  pass  all  cross-sections  of  the  stream 
during  any  given  time.  But  the  mass  passing  a  section  F 
where  the  velocity  is  v  and  the  density  w  is  wFv  per  unit  time. 
Hence,  comparing  different  sections, 


=  wFv  =  constant  .....     (1) 

A3.  Energy  Passing  a  Given  Cross-section.—  The  expression 
deduced  in  Art.  59  for  the  energy  passing  a  given  section  of 
the  stream  is  not  changed  by  the  fact  that  the  fluid  is  com- 
pressible. But  in  applying  it  to  different  sections  the  appro- 
priate value  of  w  must  be  used  at  each  section. 

*  In  other  words,  there  is  a  transformation  of  molecular  energy  into 
mechanical  or  the  reverse. 

249 


250 


STEADY  FLOW  OF  A  GAS. 


A  4.  Energy  Generated  within  a  Given  Portion  of  the  Stream 
by  Expansion. — Let  Fig.  Al  represent  a  portion  of  a  steady 
stream  of  gas,  and  let  it  be  assumed  that  the  density  decreases 
continuously  in  the  direction  of  flow.*  Let  A  and  B  be  any 
two  cross-sections,  and  consider  the  motion  of  the  body  of  gas 
AB  during  a  short  time  At,  at  the  end  of  which  it  occupies  the 
volume  A'B' '.  If  W  denotes  the  mass  of  fluid  which  passes  any 
section  per  unit  time,  we  have  for  the  mass  passing  a  section 
during  the  time  At 

WAt  =  WiFiViM  =  w2F2v2At ; 

each  of  these  expressions  being  equal  to  the  mass  of  fluid  in 
each  of  the  volumes  A  A ',  BB' '.  During  the  small  motion  con- 
sidered, each  elementary  portion 
of  gas  expands  slightly,  doing 
work  against  pressure;  it  is 
desired  to  determine  the  total 
amount  of  such  work  done  dur- 
ing the  time  At  by  all  the  ele- 
mentary portions  of  the  body 
AB.  Now  in  steady  flow  the 

condition  at  any  given  point  of 

the  volume  A'B  must  be  the 
same  at  the  end  of  the  time  At  as  at  the  beginning.  The  whole 
work  of  expansion  done  by  the  body  AB  is  therefore  equal  to 
that  which  would  be  done  by  the  elementary  portion  AA'  if 
it  expanded  into  the  volume  BB' ',  passing  through  all  stages 
as  to  temperature,  pressure,  and  density  that  actually  exist  in 
the  stream  from  A  to  B.  This  work  cannot,  therefore,  be  com- 
puted, unless  the  condition  of  the  gas  is  known  at  every  point 
in  the  stream  AB. 

Isothermal  change  of  perfect  gas.— If  the  temperature  of  the 
gas  is  the  same  at  every  point  and  remains  unchanged  during 
the  flow,  the  relation  between  pressure  and  density  for  the  case 
of  isothermal  expansion  is  to  be  used  in  computing  the  work. 

*  This  is  merely  for  convenience  in  stating  the  argument.  The  conclusion 
holds  algebraically  in  any  case. 


FIG.  Al. 


WORK  OF  EXPANSION.  251 

For  a  perfect  gas  this  relation  is 

£1-!*,  (2) 

p2      W2 

or  —  =  —  =  —  =  constant (3) 

w     w\    w2 

The  work  done  by  unit  mass  in  expanding  from  pressure  p\  and 
density  w\  to  pressure  p2  and  density  w2  is  easily  found  to  be 


The  work  of  expansion  within  the  volume  AB  per  unit  time 
is  therefore 

TF^log^.  (4) 

wi     5  p2 

Adidbatic  change  of  perfect  gas. — If  no  heat  is  given  out  or 
received  by  any  portion  of  the  gas  during  the  flow,  the  condi- 
tion of  steady  flow  must  be  such  that  the  temperature  varies 
along  the  stream  with  the  density  in  just  the  same  way  as  it 
varies  in  a  given  portion  of  gas  expanding  or  contracting 
adiabatically.  For  perfect  gases  the  adiabatic  law  connecting 
pressure  and  density  is 


M'2 


in  which  k  is  the  ratio  of  the  specific  heat  at  constant  pressure 
to  that  at  constant  volume,  and  has  the  value  1.41  very  nearly. 
The  work  done  by  unit  mass  in  expanding  from  pressure  pi 
and  density  w\  to  pressure  p2  and  density  w2  is  easily  shown 
to  be 


_ 

k  —  l\Wi     w2 

Hence  the  work  of  expansion  per  unit  time  within  the  volume 
ABis 


w 


252  STEADY   FLOW    OF   A   GAS. 

Expansion  of  steam.  —  For  saturated  steam  expanding  adi- 
abatically  the  result  given  for  perfect  gases  holds  approxi- 
mately, except  that  a  different  number  takes  the  place  of  k. 
Assuming  the  equation 


the  work  of  expansion  within  the  volume  AB  per  unit  time  is 


in  which  the  value  of  m  is  about  1.13  for  dry  saturated  steam 
and  somewhat  less  for  steam  mixed  with  water.  A  value  com- 
monly used  is  -17°-. 

The  cases  of  isothermal  and  adiabatic  change  are  the  two 
extremes  between  which  any  actual  case  of  flow  of  a  gas 
will  lie.  In  order  to  maintain  the  isothermal  condition  there 
must  be  free  conduction  of  heat  between  the  stream  and  sur- 
rounding bodies,  which  are  kept  at  constant  temperature  „ 
Practically,  the  isothermal  condition  could  be  approximated 
to  only  in  case  of  very  slow  flow.  If  the  flow  is  very  rapid,  the 
quantity  of  heat  gained  or  lost  by  conduction  by  any  portion 
of  the  gas  may  be  a  small  fraction  of  that  absorbed  or  generated 
by  reason  of  the  work  of  expansion  or  contraction,  so  that  the 
condition  of  adiabatic  change  may  be  nearly  realized. 

In  any  case,  let  W-H"  represent  the  mechanical  energy  gen- 
erated per  unit  time  within  the  volume  AB  by  reason  of  expan- 
sion. Then  for  isothermal  flow 

#»  =  alog&  .    .    (9) 

Wi       &  p2 

For  adiabatic  flow 

fl«       !(£>_£>).  .  (10) 

k-l\Wi     w2/ 


GENERAL  EQUATION  OF  ENERGY         253 

A  5.  General  Equation  of  Energy  for  Steady  Flow  of  a  Gas. 

—  Recurring  now  to  the  reasoning  used  in  Art.  60,  consider  the 
gains  and  losses  of  mechanical  energy  in  the  volume  AB  (Fig. 
A  1)  per  unit  time.  The  volume  receives  across  the  section  A 
the  amount  of  energy 


It  loses  across  the  section  B  the  amount 


It  loses  by  dissipation  an  amount  which  may  be  called  WH'. 
And  finally  it  gains  an  amount  WH"  due  to  the  work  done 
by  the  gas  in  expanding.  But  the  total  energy  gained  must 
equal  the  total  energy  lost,  since  in  steady  flow  the  quantity 
of  mechanical  energy  within  the  volume  AB  remains  constant. 
Hence 


or  Hi-H2  =  H'-H",     .....    (12) 

in  which  HI  and  H2  have  meanings  as  in  Art.  62. 

A  6.  Equation  of  Energy  for  Isothermal  Flow.  —  Using  equa- 
tion (9),  and  remembering  that  in  isothermal  change  of  volume 
PI/WI  =p2/w2,  the  general  equation  of  energy  becomes 


+ff'.    (13) 


A  7.  Equation  of  Energy  for  Adiabatic  Flow.  —  In  this  case 
equation  (10)  is  to  be  used,  and  the  general  equation  of  energy 
may  be  written  in  the  form 


254  STEADY    FLOW   OF    A    GAS. 

In  applying  this  equation  it  must  be  remembered  that  p  and  w 
are  connected  by  the  relation  (5). 

Equation  (14)  applies  approximately  to  steam,  k  being  given 
the  proper  value. 

A  8.  Flow  of  Gas  Through   Small   Orifice.  — If  a  chamber 
(X,  Fig.  A  2)  containing  gas  under  pressure  be  connected  by  a 

small  orifice  or  short  tube  with  a 
second  chamber  (Y)  in  which  a  less 
pressure  exists,  flow  will  take  place 
under  practically  adiabatic  condi- 
tions. Neglecting  loss  of  energy  by 
dissipation,  a  formula  for  the  veloc- 


FIG  A  2  ity  of  flow  through  the  orifice  may 

P&  be  deduced  from  equation  (14). 

Let  p'  be  the  pressure  within  the  chamber  X,  p"  that  within 
the  chamber  Y,  and  p0  that  within  the  stream  passing  through 
the  orifice.*  In  applying  the  equation  of  energy,  let  the  point 
corresponding  to  A,  Fig.  A  1,  be  taken  within  the  chamber  X, 
and  the  point  corresponding  to  B  at  the  smallest  cross-section 
of  the  stream  passing  the  orifice.  The  terms  z\  and  z2  may  be 
neglected,  and  we  have  also 

Vi=0,     PI/W!=P'/W',    V2  =  v0, 
so  that  the  equation  reduces  to 


__ 

2g~k-l\w'      w 

Or,  making  use  of  the  relation  between  pressures  and  densities 
in  adiabatic  change,  we  may  write 

»o2      k 


The  velocity  of  flow  through  the  orifice  thus  depends  upon 
the  ratio  of  the  pressure  within  the  orifice  to  that  within  the 
chamber  X. 

*  It  might  seem  that  p0  could  be  assumed  equal  to  p",  but  it  will  soon 
appear  that  this  is  not  generally  permissible. 


FLOW  OF  GAS  THROUGH   SMALL  ORIFICE. 


255 


Mass  discharged  per  unit  time. — If  the  cross-section  of  the 
jet  is  FQ,  we  have  for  the  mass  discharged  per  unit  time 


W  =  wQF0v0  =     j 


It  will  be  seen  that  if  p0/p'  be  assumed  to  decrease  from  the 
value  1,  W  will  increase  up  to  a  maximum  and  then  decrease. 
The  maximum  occurs  when 


+  l 


(18) 


It  appears,  therefore,  that  po/p'  will  not  fall  below  this  value, 
however  small  the  ratio  p"/p'.  Substituting  this  limiting  value 
of  PQ/P'  in  (16)  and  (17),  the  former  becomes 


k      pr 


2g    k  +  1  w" 
and  the  latter  takes  the  form 


(19) 


5-7TTT.      •     .     •     (20) 


The  limiting  values  of  po/p'  for  two  cases,  with  the  corre- 
sponding formulas  for  VQ  and  W,  are  as  follows : 


Substance. 

k 

Limiting 
Po/p'- 

Formula  for  VQ. 

Formula  for  W. 

f 

1     rf 

Air 

1  41 

.527 

V  =  7Q5\*2o 

W-  485w'Fn  v  |2<7— 

\     w' 

Steam.  .  . 

1.111 

.582 

<^£& 

W-.  446^^2^ 

256 


STEADY    FLOW   OF   A    GAS. 


The  value  of  j//v/  for  air  at  0°  C.  is  26,200  ft.,  while  at 
t°  -C.  it  is  26,200(1 +  .003660.  For  any  given  temperature, 
therefore,  the  velocity  as  given  by  (19)  is  independent  of  the 
value  of  p'.  For  0°  C.  it  is  v0  =  990  ft.  per  sec. 

For  steam  at  an  absolute  pressure  of  100  Ibs.  per  sq.  in.  the 
value  of  p' '/w'  is  about  63,300  ft.;  at  200  Ibs.  per  sq.  in.  it  is 
about  66,200  ft.  The  corresponding  values  of  the  velocity  of 
efflux  as  given  by  (19)  are  1,460  ft.  per  sec.  and  1,500  ft.  per  sec. 

These  results  do  not  apply  if  p"  is  greater  than  the  value  of 
po  given  by  (18);  in  that  case  equations  (16)  and  (17)  must 
be  used,  with  po  =  p". 

A  9.  Complete  Solution  of  Problem  of  Adiabatic  Flow.— In 
the  case  of  flow  considered  above,  in  which  p" <po,  if  steady 

flow  continues  beyond  the  section 
at  which  the  pressure  is  PQ,  it  may 
be  shown  that  both  the  cross-section 
and  the  velocity  of  the  stream  will 
increase  as  the  pressure  decreases. 
This  is  brought  out  in  the  following 
complete  solution  of  the  problem  of 
adiabatic  flow. 

Let  p  be  the  pressure  and  v  the  velocity  at  a  point  where 
the  cross-section  is  F;  then  the  relations  between  these 
three  quantities  are  expressed  by  equations  similar  to  (16) 
and  (17)  • 


X.\J?^Q* 

P' 

SQi  Y 

FIG.  A  3. 

p' 


(21) 


By  assuming  values  of  p/p'  the  corresponding  values  of  F 
and  v  can  be  computed  from  these  equations.    Or,  using  equa- 


SOLUTION    OF    PROBLEM    OF    ADIABATIC    FLOW.         257 


tions(19)  and  (20),  the  relations  may  be  expressed  in  the  fol- 
lowing convenient  forms  : 


,24) 
V  2  /       \j/J     VQ 

Application  to  air.  —  Putting  A;  =  1.41,  these  equations  become 

11 

J;       ....    (25) 


291 


The  accompanying  table  gives  values  of  v/v0  and  F/F0  com- 
puted from  these  equations,  together  with  the  corresponding 
values  of  w/u/.  The  same  results  are  shown  graphically  in 
Fig.  A  4 


p 

pf 

w 
w' 

V 
VQ 

F 

Fo 

p 

P'  . 

w 
w' 

V 

VQ 

F 

F0 

0 

0 

2.43 

00 

.526 

.634 

1.00 

1.00 

.05 

.119 

.85 

2.87 

.60 

.696 

.903 

1.01 

.10 

.195 

.70 

.91 

.70 

.776 

.762 

1.07 

.15 

.260 

.58 

.54 

.80 

.853 

.607 

1.22 

.20 

.318 

.48 

.32 

.85 

.891 

.521 

1.37 

.30 

.426 

.32 

.13 

.90 

.928 

.    .422 

1.61 

.40 

.522 

.17 

.03 

.95 

.964 

.296 

2.22 

.50 

.614 

1.04 

1.01 

1.00 

1.000 

0 

00 

Application  to  steam. — The  equations  for  steam,  obtained  by 
putting  k=  V,  are  as  follows: 


258 


STEADY    FLOW    OF    A   GAS. 


Values  computed  from  these  equations  are  given  in  the  follow- 
ing table,  and  the  corresponding  curves  are  shown  in  Fig.  A  4. 


p 

p' 

w 
w' 

V 

Vo 

F 

FQ 

p 
P' 

w 
w' 

V 

v^ 

F 

Fo 

0 

0 

4.37 

00 

.582 

.614 

1.00 

.00 

.05 

.067 

2.22 

4.10 

.60 

.631 

.973 

.00 

.10 

.126 

1.97 

2.47 

.70 

.725 

.817 

.04 

.15 

.181 

.81 

1.87 

.80 

.818 

.648 

.16 

.20 

.235 

.68 

1.56 

.85 

.863 

.555 

.28 

.30 

.338 

.47 

1.24 

.90 

.909 

.447 

.51 

.40 

.438 

.29 

1.09 

.95 

.955 

.311 

2.07 

.50 

.536 

.13 

1.02 

1.00 

1.000 

0 

00 

The  above  numerical  results  may  be  applied  to  any  case  of 
adiabatic  flow  of  air  or  steam,  if  the  pressure  has  known  values 
p'  and  p"  at  any  two  points.  The  two  cases  p"  <po  and  p" >  po 
require  different  treatment,  as  will  be  seen  from  the  following 
examples. 

EXAMPLES. 

1.  A  reservoir  contains  air  at  20°  C.  under  a  pressure  of  20  Ibs.  per 
sq.  in.  (absolute).  Compute  the  mass  discharged  per  second  into  the 
atmosphere  through  an  orifice  of  one  square  inch  area. 

The  rate  of  mass-discharge  is  W  =wFv.  In  this  case  p">p,}  we 
therefore  take  p"  as  the  pressure  at  the  orifice,  and  find  the  correspond- 
ing values  of  w  and  v.  For  20°  C.  we  have  p' /w' =  26200(1  +  .0732)  = 
28100  ft.,  and  since  p'  =2880' Ibs.  per  sq.  ft.,  w'  =.01025  Ibs.  per  cu.  ft. 
From  the  above  table  or  diagram,  w/w'=.8Q5,  hence  w  =  . 00825.  Also 
^  =  .705;  Vj  =  .765 V64.4  X  28100  =  1030;  v  =  . 705X1030  =726  ft.  per 


sec. 


Finally,  W  =  wFv  =  . 00825  X— X 726 


.0416  Ibs.  per  sec. 


2.  Air  is  discharged  from  a  reservoir  in  which  the  pressure  is  100 
Ibs.  per  sq.  in.  above  atmospheric,  and  the  temperature  20°  C.,  through 
an  orifice  of  one  square  inch  area.  Compute  the  mass  discharged  per 
second. 

In  this  case  p"<p0,  so  that  formulas  (19)  and  (20)  apply  directly. 
The  value  of  p' /w'  is  28100  ft.,  as  in  Ex.  1;  p' =  114.7  X 144  =  16520  ; 
w'  =  16520  /28100  =  .  588;  F0  =  1/144.  Hence  the  formulas  give 

.485  X. 588  X-*- 
144 


.  765^64.4X28100  =  1030      ft.     per     sec.;       W 


X  V64.4X28100=2.66  Ibs.  per  sec. 


SOLUTION    OF    PROBLEM    OF    ADIABATIC    FLOW.         259 


2.5 


2.0 

£ 

fe 


Adiabatic  flow  of  gas. 


Air 


\ 


1.0 


^^ 


.1 


.3         .4          .5          .6         .7         .8          .9        1.0 

Values  of 

FIG.  A  4. 


260  STEADY    FLOW    OF   A    GAS. 

3.  In  Ex.  1,  if  the  discharge  is  through  a  tube  whose  smallest  cross- 
section  is  one  square  inch  and  which  diverges  beyond  this  section,  deter- 
mine the  pressure  and  velocity  at  a  section  whose  area  is  1.2  square 
inches. 

Here  F/FQ  =  1.2,  and  Fig.  A  4  gives  p/p'=.262,  v/v0  =  1.38;  hence 
p=30.0  Ibs.  per  sq.  in.,  v  =  l42Q  ft.  per  sec. 

4.  With  conditions  as  in  Ex.  3,  what  value  of  the  cross-section  of  the 
tube  would  bring  the  pressure  down  to  14.7  Ibs.  per  sq.  in.?     Determine 
the  corresponding  velocity. 

Here  p/p'  =  14.7/114.7  =  .128,  F/F0  =  1.7,  v/v0  =  1.63.  Hence 
F  =  1.7  sq.  in.,  v  =  1680  ft.  per  sec. 

5.  With  conditions  as  in  Ex.  3  except  that  the  discharge  is  into  a 
vacuum,  what  greatest  velocity  would  be  acquired? 

Ans.  If  p=0,  v  =2A3v0  =2500  ft.  per  sec. 

6.  Steam  under  a  pressure  of  100  Ibs.  per  sq.  in.  (above  atmosphere) 
is  discharged  into  the  air  through  an  orifice  one  square  inch  in  area. 
Compute  the  mass  discharged  per  second. 

Since  p"  < p0,  formulas  (19)  and  (20)  apply  with  k  =  10/9.  From  steam 
tables  we  find  w/  =  .2580  Ibs.  per  cu.  ft.,  hence  p'/w'  =  114.7X144/.2580 
=  64100  ft.;  v0  =  1480  ft.  per  sec.;  W  =  1.63  Ibs.  per  sec. 

7.  With  data  as  in  Ex.  6,  if  the  discharge  is  through  a  diverging 
tube,  compute  the  pressure  and  velocity  at  a  point  where  the  cross- 
section  is  20  per  cent  greater  than  the  smallest  section. 

Ans.  v=2110  ft.  per  sec.;  p=36.7  Ibs.  per  sq.  in.  (absolute). 

8.  With  conditions  as  in  Ex.  7,  what  value  of  F/F0  would  bring  the 
pressure  down  to  14.7  Ibs.  per  sq.  in.?     Determine  the  corresponding 
velocity.  Ans.  F/FQ  =2.2;    v  =2780  ft.  per  sec. 


APPENDIX  B. 
RELATIVE  MOTION. 

Bl.  Meaning  of  Relative  Motion. — In  the  theory  of  tur- 
bines we  are  concerned  with  the  motion  of  a  particle  of  water 
with  respect  to  the  earth,  and  also  with  its  motion  with  respect 
to  the  rotating  wheel.  In  order  to  understand  just  what  is 
meant  by  these  two  motions,  consider  the  following  illustration. 

Conceive  a  horizontal  platform  to  be  rotating  uniformly 
(with  respect  to  the  earth)  about  a  fixed  vertical  axis,  and 
suppose  an  observer  stationed  upon  the  platform,  to  whom  the 
earth  is  invisible  and  inaccessible.  He  will  regard  the  plat- 
form as  at  rest,  no  other  body  being  available  with  which  he  can 
compare  its  motion.  A  body  moving  on  the  platform  would 
trace  upon  it  a  certain  line  which  to  the  obseiver  would  be  its 
true  path.  At  any  instant  it  would  appear  to  be  moving  in 
this  path  in  a  certain  direction  at  a  certain  rate,  and  to  the 
observer  this  would  be  its  true  velocity.  If  the  body  were 
visible  to  a  second  observer  stationed  upon  the  earth,  it  would 
to  him  appear  to  follow  quite  a  different  path,  and  at  any  in- 
stant it  would  appear  to  be  moving  at  a  different  rate  and  in  a 
different  direction. 

By  the  motion  relative  to  the  platform  would  be  meant  the 
motion  as  estimated  by  the  observer  on  the  platform,  while 
the  motion  relative  to  the  earth  would  mean  the  motion  as  esti- 
mated by  the  observer  on  the  earth. 

Similarly,  in  case  of  water  flowing  through  a  rotating  wheel, 
the  motion  of  a  particle  relative  to  the  wheel  means  its  motion 
as  it  would  be  estimated  by  a  person  to  whom  the  wheel  appeared 
stationary. 

261 


262 


RELATIVE    MOTION. 


B  2.  Absolute  and  Relative  Motion. — In  the  theory  of  tur- 
bines, as  in  all  ordinary  practical  problems,  the  earth  is  regarded 
as  a  fixed  body,  and  motion  with  respect  to  it  is  for  convenience 
called  "  absolute,"  while  motion  with  respect  to  the  rotating 
wheel  is  called  " relative." 

It  is  necessary  to  consider  definitely  the  relation  between 
these  two  motions. 

B3.  Relation  between  Absolute  and  Relative  Velocities  of 
a  Particle. — Let  the  wheel  rotate  with  uniform  angular  velocity 
oj  about  an  axis  represented  in  Fig.  B  1  by  the  point  0,  the  axis 


o  «e=-~jir_i:L T 


being  perpendicular  to  the  plane  of  the  figure.  Let  AB  be  the 
path  traced  by  a  particle  upon  the  rotating  wheel  during  a 
time  At.  Every  point  of  the  wheel  describes  a  circle  about  the 
axis  0;  the  point  initially  at  A  describes  during  At  an  arc  AA', 
and  the  point  initially  at  B  an  arc  BBf  ',  the  angles  AOAr,  BOB' 
being  equal.  The  moving  particle  describes  with  respect  to 
the  earth  a  path  from  A  to  B'  . 

Let  V  denote  the  absolute  velocity  of  the  particle  when  at 
A,  v  its  relative  velocity  when  at  A,  and  u  the  absolute  velocity 
of  the  point  A  of  the  wheel.  These  velocities  are  directed  along 
the  tangents  to  the  curves  AB'  ',  AB,  AA',  respectively,  at  the 
point  A.  Their  vector  values  are  * 

vector  AB' 


*  Theoretical  Mechanics,  Art.  244.    The  brackets  are  used  to  denote 
vector  values,  as  explained  in  Art.  174. 


ABSOLUTE   AND    RELATIVE    VELOCITIES.  263 

vector  AB 
~JT 

vector  AAf 


[v]  =limit(^ 
[u]  =limitu,=0) 


In  the  limit  the  figure  ABB-A'  is  a  parallelogram,  and 
vector  AB'  =  vector  AB  +  vector  A  A', 

hence  IT]=M+N- 

That  is,  in  words, 

The  absolute  velocity  of  a  particle  is  equal  to  the  vector  sum 
of  its  relative  velocity  and  the  absolute  velocity  of  that  point  of  the 
wheel  which  momentarily  coincides  with  the  particle. 

Notice  that  the  reasoning  holds  even  if  A  B  and  ABf  are 
not  in  a  plane  perpendicular  to  the  axis,  so  that  they  are  not 
shown  in  true  size  in  the  figure. 

B4.  Relation  between  Absolute  and  Relative  Accelerations. 

— Space  will  not  be  taken  to  deduce  the  relation  between  abso- 
lute and  relative  accelerations,  since  it  is  not  required  in  the 
theory  of  turbines  as  given  in  the  text.  It  is  discussed  in  the 
author's  Theoretical  Mechanics,  Chapter  XXIV. 

EXAMPLES. 

1.  A  wheel  rotates  uniformly  at  the  rate  of  120  R.P.M.     A  particle 
4  ft.  from  the  axis  has  a  velocity  relative  to  the  wheel  of  30  ft.  per  sec. 
directed  at  an  angle  of  45°  to  the  perpendicular  from  the  particle  to  the, 
axis  of  rotation.     Compute  its  velocity  relative  to  the  earth. 

Ans.  74.6  ft.  per  sec.  directed  at  angle  73°  30'  to  the  radial  line. 

2.  A  wheel  rotates  uniformly  at  the  rate  of  120  R.P.M.     A  particle 
4  ft.  from  the  axis  has  an  absolute  velocity  of  30  ft.  per  sec.  directed 
at  an  angle  of  45°  to  the  perpendicular  from  the  particle  to  the  axis  of 
rotation.    Compute  its  relative  velocity. 

Ans.  36.0  ft.  per  sec.  directed  at  angle  53°  50'  to  the  radial  line. 


APPENDIX  C. 
CONVERSION  FACTORS. 

THE  following  table,  designed  to  facilitate  the  reduction 
from  one  unit  or  system  of  units  to  another,  includes  only  such 
factors  as  are  likely  to  be  of  use  in  the  problems  of  practical 
Hydraulics.  Five-figure  values  are  in  most  cases  given,  although 
hydraulic  measurements  do  not  often  warrant  this  degree  of 
accuracy. 

The  numbers  which,  involve  the  density  of  water  are  based 
upon  the  value  for  pure  water  at  maximum  density.  The  varia- 
tion in  density  by  reason  of  impurities  (except  in  the  case  of 
sea-water  or  water  in  salt  lakes)  is  ordinarily  inappreciable, 
and  the  correction  for  temperature  can  be  applied  by  aid  of 
the  table  in  Art.  8. 

The  ratio  of  the  density  of  mercury  to  that  of  water  has 
been  taken  as  13.596. 


1  inch 
1  foot 
1  mile 
1  centimeter 

1  kilometer 


LENGTH. 

2.5400 
30.479 
1.6093 
0.39371 
0.032809 
0.62138 


centimeters 

centimeters 

kilometers 

inch 

foot 

mile 


Log. 

.40483 
1.48401 

.20664 
1.59517 
2.51599 
1.79336 


1  square  inch 
1  square  foot 
1  square  centimeter 
1  square  meter 


AREA. 

6.45137 

928.997 

0.15501 

10,76430 

264 


square  centimeters  .  80965 

square  centimeters  2.96801 

square  inch  1 . 19035 

square  feet  1.03199 


CONVERSION    FACTORS. 


265 


VOLUME. 


Log. 

1  cubic  inch 

=       16.386 

cubic  centimeters 

1.21448 

1  cubic  foot 

0.028315 

cubic  meter 

2.45202 

=       28.315 

liters 

1.45202 

7.4805 

U.  S.  gallons 

.87393 

=         6.2321 

British  Imperial  gallons 

.79463 

1  cubic  centimeter 

0.061027 

cubic  inch 

2.7855-2 

1  liter  (cubic  decimeter) 

=       61.027 

cubic  inches 

1.78552 

0.035317 

cubic  foot 

2.54798 

0.26419 

U.  S.  gallons 

1.42191 

1  cubic  meter 

-       35.317 

cubic  feet 

1.54798 

1  U.  S.  gallon 

=     231. 

cubic  inches 

2.36361 

•=*         0.13368 

cubic  foot 

1.12607 

3.78521 

liters 

.57809 

0.83311 

British  Imperial  gallon 

1.92070 

1  British  Imperial  gallon 

=     277.27 

cubic  inches 

2.44291 

0.16046 

cubic  foot 

1.20537 

«=         4.5435 

liters 

.65739 

1.2003 

U.  S.  gallons 

.07930 

MASS  AND  WEIGHT. 

1  pound 

0.45359 

kilogram 

1.65667 

1  kilogram 

=         2.2046 

pounds 

.34333 

WEIGHT  OF  WATER 

(AT  4°  CENT.). 

1  cubic  foot  weighs 

62.424 

pounds 

1.79535 

28.315 

kilograms 

1.45202 

1  U.  S.  gallon  weighs 

8.3448 

pounds 

.92142 

3.7852 

kilograms 

.57810 

1  Brit.  Imp.  gal.  weighs 

10.0165 

pounds 

1.00072 

4.5435 

kilograms 

0.65739 

1  cubic  meter  weighs 

1000. 

kilograms 

3.00000 

2204.7 

pounds 

4.34333 

VELOCITY 

, 

1  foot  per  second 

0.68182 

miles  per  hour 

1.83367 

1  mile  per  hour 

1.4667 

feet  per  second 

0.  16633 

RATE  OF  DISCHARGE. 

1  cubic  foot  per  second 

7.4805 

U.  S.  gallons  per  second 

.87393 

=     448.83 

U.  S.  gallons  per  minute 

2.65208 

=         6  .  2321 

B.I.  gallons  per  second 

.79463 

=     373.93 

B.  I.  gallons  per  minute 

2.57278 

1  U.  S.  gallon  per  second 

=         0.13368 

cubic  foot  per  second 

1.12607 

266 


CONVERSION    FACTORS. 


RATE  OF  DISCHARGE— Continued. 


1  U.  S.  gallon  per  minute  = 
1  million  U.  S.  gallons  per 

day  = 

1  B.  I.  gallon  per  second  = 
1  B.  I.  gallon  per  minute  = 
1  million  B.  I.  gallons  per 

day  = 


Log. 


1  pound  per  square  inch 


1  pound  per  square  foot 


1  kilogram  per  sq.  meter  = 


1  atmosphere 


0 . 0022280  cubic  foot  per  second  3 . 34792 

1.5472        cubic  feet  per  second  .18956 

0 . 16046       cubic  foot  per  second  1 . 20537 

0 . 0026743  cubic  foot  per  second  3 . 42722 

1 . 8572        cubic  feet  per  second  .  26886 


PRESSURE. 


144 .  pounds  per  square  foot 

70 . 3 1 0  grams  per  sq .  centimeter 

2 . 3068  feet  water  column 

2.0360  inches  mercury  column 

4 . 8826  kilograms  per  sq .  meter 

0.016020  feet  water  column 

0.014139  inches  mercury  column 

0 . 20480  pounds  per  square  foot 

'         0.1  centimeter  water  column 

0 . 0073552  centimeter  mercury  col- 
umn 

30 . 00  inches  mercury  column 

34 . 00  feet  water  column 

2122 .  pounds  per  square  foot 

14.73  pounds  per  square  inch 

76.00  centimeters      mercury 

column 

10 . 33  meters  water  column 

10330.  kilograms  per  sq.  meter 

1033.  grams  per  sq.  centimeter 


2.15836 

1 . 84702 

.36301 

.30878 

.68866 

2.20465 

2.15042 

1.31134 

1.00000 

3.86659 
1.4771 
1.5315 
3.3268 
1 . 1685 


0142 
0142 


3.0142 


INDEX. 


INDEX. 


Adiabatic  change  of  perfect  gas,  251 

expansion  of  steam,  252 

flow  of  gas,  complete  solution,  256 
Admission,  complete  and  partial,  186 
Air,  effect  of,  on  flow  in  pipes,  97 
Angular  momentum,  principle  of,  168 
Approach,  velocity  of,  55,  56, 149, 150 
Atmospheric  pressure,  11 
Backwater,  139 

Balancing  of  turbine  pump,  247 
Bazin's  formula  for  open  channels, 

126 

Bend,  loss  of  head  due  to,  75 
Bernoulli's  theorem,  47 
Bourdon  gauge,  68 
Branching  pipe,  95 
Canal  lock,  42 

Capillary  tubes,  loss  of  head  in,  80 
Center  of  pressure,  12,  13 
Centrifugal  pump,  224 

elementary  illustration  of,  224-227 
Chezy's  formula  for  long  pipe,  99,  109 

for  open  channel,  124 

variation  of  coefficient  in,  125 
Coefficient  of  contraction,  34 

of  discharge,  35 

of  velocity,  34 

Compounding  of  turbine  pumps,  247 
Compressibility,  its  effect  on  theory 
of  steady  flow,  249 

of  water,  5 
Confined  stream,  action  of ,  upon  pipe, 

169 
Contraction ,  coefficient  of,  34 

loss  of  head  due  to,  74 
Conversion  factors,  264 
Critical  velocity,  112 
Current-meter,  144 
Dam,  pressure  on,  19 

submerged,  155 
Darcy's  formula  for  friction  loss,  72, 

100,  108, 113 
Density  of  water,  5 
Diffuser,  219 
Discharge,  coefficient  of,  35 

rate  of,  30 
Dynamic  action  of  streams,  160-175 


Effective  head,  49 

Efficiency  of  turbine  motor,  189 

of  turbine  pump,  248 
Elasticity,  4 

Emptying  a  reservoir,  40 
Energy,  available,  188 

equation     of,     for    flow     through 

rotating  wheel,  181 
for  large  stream,  116-121 
for  open  stream  of  variable  sec- 
tion, 135 

for  steady  flow  of  gas,  253 
for  steady  flow  of  liquid,  47 
with  losses,  64-84 
without  losses,  52-63 
with  pump  or  motor,  85-89 
given  up  to  wheel,  180 
passing  a  cross-section,  45 
theory  of,  applied  to  steady  flow 

of  gas,  249-260 
applied  to  steady  flow  of  liquid, 

43-51 
transformation  and  transference  of 

43 

Enlargement,  loss  of  head  due  to,  73 
Entrance  loss  of  head,  72 
Equation  of  continuity  for  compres- 
sible fluid,  249 
for  incompressible  fluid,  31 
Exponential  formula  for  friction  loss, 

110 

Filling  a  reservoir,  41 
Floating  body,  equilibrium  of,  23-26 
Floats,  measurement  of  velocity  by 

143 

Fluid  friction,  laws  of,  107 
Forces,  surface  and  bodily,  5 
Fourneyron  turbine,  187 
Francis,  weir  formula  of,  149 
Francis  turbine,  188,  208,  209 
Friction  coefficients,  112-115 
Friction  loss,  formulas  for,  104-113 

theory  of,  105 

Gas,  steady  flow  of,  249-260 
Girard  turbine,  187 

with  axial  flow,  theory  of,  196,  197 
with  radial  flow,  theory  of,  191-195 


270 


INDEX. 


Gravel,  flow  through,  81 
Head,  48,  49 
Head,  effective,  49 
Hydraulic  gradient,  76,  83 
relation  of  pipe  to,  96 

slope,  77 

Hydraulics  denned,  1 
Ideal  velocity  and  discharge,  34 
Impulse  turbine  defined,  186 

theory  of,  190-198 
Isothermal  change  of  perfect  gas,  250 

flow  of  gas,  equation  of  energy  for, 

253 
Jet,  force  causing  deflection  of,  164 

force  producing,  161 

reaction  of,  162 

striking  fixed  surface  normally,  162 

striking  moving  surface  normally, 

162 
Jet  water-wheel  with  flat  vanes,  163 

with  curved  vanes   167 
Jonval  turbine,  188 
Kutter's  formula,  109,  127 

graphical    representation    of,    128, 
132 

reliability  of,  128 

table  computed  from,  129 
Large  pipes,  friction  factors  for,  114, 

115 

Level  surface,  10 
Long  pipe,  99 
Loss  of  head,  meaning  of,  50 

at  entrance  to  pipe.  72 

due  to  bend,  75 

due  to  contraction,  74 

due  to  obstruction,  74 

due  to  sudden  enlargement,  73   " 

experimental  determination  of,  64, 
102 

in  capillary  tubes,  80 

in  open  channel,  135 

in  short  tube,  70 

in  standard  orifice,  69 

in  uniform  pipe,  71,  72 

methods  of  estimating,  64 
Masonry  dam,  pressure  on,  19 
Mercury  gauge,  67 
Metacenter,  25 
Miner's  inch,  148 

Minimum  cross-section  for  given  dis- 
charge, 131 

Momentum,  principle  of,  160 
Motor,  equation  of  energy  for,  85,  86 
Obstruction,  loss  of  head  due  to,  74 
Open  channels,  non-uniform  flow  in, 
135-141 

uniform  flow  in.  122-134 
Orifice,  flow  through.  34.  53   54 

flow  of  gas  through,  254 


Orifice,  large,  35,   36 

rectangular,  36 

standard,  35 

submerged,  36 
Orifices,    measurement    of    rate    of 

discharge  by,  145-148 
Piezometer,  water,  65 
Pipe  leading  from   reservoir    57-60 

90,  99 

Pipe,  loss  of  head  in,  71,  72,  102-115 
Pipes,  long,  99 
Pi  tot's  tube,  144 

theory  of,  171 

Pressure  at  any  section  of  pipe,  58 
Pressure  gauge,  Bourdon,  68 

mercury,  67 

water,  65 

Pressure  in  fluid  acted  on  by  gravity, 
9 

in  fluid  free  from  bodily  forces,  8 

in   terms  of  height  of  liquid  col- 
umn, 10 

intensity  of,  3 

negative,  59 

on  curved  surface,  17-19,  21 

on  different  planes,  7 

on  plane  area,  12-16 
Prony's  formula  for  friction  loss,  108 
Pump,  equation  of  energy  for,  87, 
88 

turbine,  224-248 
Ram  pressure,  173 
Rate  of  discharge,  direct   measure- 
ment of,  142,  143 

indirect  measurement  of,  145 
Reaction  turbine,  definition  of,  186 

different  cases  of  flow  in,  218 

discharging    into    diverging    pas- 
sages, 219 

efficiency  of,  221 

general  features  of,  208 

general  theory  of,  210-^217 

regulation  and  governing  of,  221 
Rectangular  orifice,  36 
Rectangular  weir,  37 
Relative  motion,  261-263 
Reservoir,  emptying  and  filling  of,  40, 

41 

Resultant  pressure,  12,  13 
Reversal  of  stream  by  fixed  vane,  165 

by  moving  vane,  166 
Rotating  liquid,  form  of  free  surface 
of,  28 

variation  of  pressure  in,  26 
Rotating  vane,  action  of  stream  on, 

168 
Rotating  wheel,    theory   of    steady 

flow  through.  176-184 
Short  tube,  loss  of  head  in,  70 


INDEX 


271 


Smith,  weir  formula  of,  150 
Solid  and  fluid  defined,  1,  2 
Stability  of  equilibrium  of  floating 

body,  24-26 
Standard  orifice,  35 

loss  of  head  in,  69 
Steady  flow  of  a  gas,  249-260 

of  water,  30 

Stress,  normal  and  tangential,  2 
Submerged  body,  pressure  on,  22 
Submerged  dam,  155 
Submerged  weir,  .153 
Suction  tube,  220 
Surface  curve,  136-140 
Tangential  water-wheel,  187, 199-207 

efficiency  of,  206 

form  of  buckets  of,  205 

general  features  of,  200 

governing  of,  207 

regulation  of,  206 

theory  of,  203-205 
Torricelli's  theorem,  32 
Triangular  weir,  37,  152 
Turbine,  definition  of,  185 
Turbine  pump,  224 


Turbine  pump,  balancing  of,  247 

case  of  constant  lift,  239-243 

case  of  constant  speed,  244-246 

compound,  247 

design  for  efficient,  228 

general  theory  of,  230-239 

lifts  and  efficiencies  of,  248 
Turbines,  classes  of,  185,  186 
Uniform  flow,  123 
Varying  head,  discharge  under,  40 
Velocity,  absolute  and  relative,  179, 
262 

coefficient  of,  34 

mean,  31 

methods  of  measuring,  143 

of  jet,  32 

Venturi  meter,  157 
Waste  weir,  151 
Weir,  rectangular,  37 

submerged,  153 

triangular,  37,  152 

with  velocity  of  approach,  56 
Weirs,  measurement  of  rate  of  dis- 
charge by. 148-152 
Wells,  flow  of  81-84 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $I.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


7946 


FEB  12 1942  R 


f  EB 


t 


PEC  14  1942 


JUN    5  1945 


:C'D  LD 


DEC  i  i  laa 

LD  21-100m-7,'40 (6936s) 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


